Global Asymptotic Stability of the Classical PID Controller by Considering Saturation Effects in Industrial Robots

Abstract

An unsolved ancient problem in position control of robot manipulators is to find a stability analysis that proves global asymptotic stability of the classical PID control in closed loop with robot manipulators. The practical evidence suggests that in fact the classical PID in industrial robots is a global regulator. The main goal of the present paper is theoretically to show why in the practice such a fact is achieved. We show that considering the natural saturations of every control stage in practical robots, the classical PID becomes a type of saturated nonlinear PID controller. In this work such a nonlinear PID controller with bounded torques for robot manipulators is proposed. This controller, unlike other saturated nonlinear PID controllers previously proposed, uses a single saturation for the three terms of the controller. Global asymptotical stability is proved via Lyapunov stability theory. Experimental results are presented in order to observe the performance of the proposed controller.

1. Introduction

Stability of PID control in robot manipulators has been subject of extensive research in past and recent years. Most industrial robots are equipped with PID controllers. It has been proved that PID controllers guarantee semiglobal asymptotical stability of the closed-loop equilibrium point in the case of set-point control [1]-[7]. Since PID controllers, until now, have only been proved to be semiglobally asymptotically stable, several nonlinear PID controllers have been proposed [8]-[10]. Unlike linear PID controllers, the nonlinear PIDs yield global asymptotical stability of the closed-loop equilibrium point. Recently, a particular case of the class of nonlinear global PID regulators originally proposed in [10] was presented in [11]. A “delayed” PID, which switched from a PD control to a PID control after a short transient, achieving global asymptotic stability was proposed in [12].

On the other hand, it is known that actuators are unable to supply unlimited torques. Therefore, several controllers taking into account this important physical limitation have been proposed in [13]-[20]. In the case of linear PID controllers with bounded torques, semiglobal asymptotical stability has been proved in [21]-[22]. Nonlinear PID controllers with bounded torques have been proposed in [23]-[26], proving global asymptotical stability. The structure of the controller presented in Theorem 1 of [26] and a similar stability analysis were previously presented in [24]. As far as the authors know, the work introduced in [23] was the first bounded PID-like controller in assuring global regulation; the work, introduced in [24], also guarantees global regulation, but with the advantage of a controller structure which is simpler than that presented in [23]. Other works involving robot tracking controllers with bounded inputs are: [27]-[33].

An unsolved ancient problem in position control of robot manipulators is to find a stability analysis that proves global asymptotic stability of the classical PID control in closed loop with robot manipulators. While this problem remains open, one can not think that the regulation problem of robot manipulators is totally solved [34]. In this sense, it is well known that industrial robot manipulators are equipped with linear PID controllers in order to deal with the position regulation problem [35]. The practical evidence suggests that in fact the classical PID in industrial robots is a global regulator. The main goal of the present paper is theoretically to show why in the practice such a fact is achieved. To this end, notice that in industrial robots (see Fig. 1) the position control is carried out via a control computer doing the job of a position outer proportional loop, which gives the command to the velocity PI servo-drives, intrinsically contained in the hardware of the motor drivers. The overall closed loop system without considering saturation effects results in a classical position PID controller [36].

Figure 1.

Scheme of the proposed nonlinear PID controller with bounded torques for robot manipulators.

However every control stage suffers saturation effects due to physical constraints of the electronic amplifiers as well as the actuators. In this case, by considering the saturation constraints in each control stage, the classical PID becomes a saturated nonlinear PID control, understanding by this, a nonlinear PID controller —like that proposed by the first time by Prof. Arimoto in [37]—, which now, in our proposed work, is overall saturated by a final saturation due to the effects of the actuator and the PI velocity servo-driver. Unlike other global nonlinear PID controllers such as those proposed in [23]-[26], the control scheme proposed in this paper uses a single saturation function for the proportional, derivative and integral terms (see Fig. 1).

A previous work of some of the authors [38] presents a PID controller with a similar structure, but concluding only local asymptotic stability. In contrast, now the proposed controller in this work, holds an industrial structure, but ensuring global asymptotic stability.

The paper is structured as follows: In Section II, we present some preliminaries; In Section III, we present the proposed controller and the stability analysis; in Section IV, we present experimental results which show the performance of the proposed controller, while Section V presents the conclusions of the paper.

Throughout this paper, we use the notation ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{A\right(x\left)\right\}$ and ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{A\right(x\left)\right\}$ , to indicate the smallest and largest eigenvalues, respectively, of a symmetric positive definite bounded matrix A(x), for any $x∊{ℝ}^n$ . Also, we define ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{A\right\}$ as the greatest lower bound (infimum) of ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{A\right(x\left)\right\}$ , for all $x∊{ℝ}^n$ . Similarly, we define ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{A\right\}$ as the least upper bound (supremum) of ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{A\right(x\left)\right\}$ for all $x∊{ℝ}^n$ . The norm of vector x is defined as $‖x‖=\sqrt{x^{T}x}$ and that of matrix A(x) is defined as the corresponding induced norm $‖A\left(x\right)‖=\sqrt{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\left(x\right)A}^{T}A\right(x\left)\right\}}$ .

2. Preliminaries

2.1 Robot dynamics

The dynamics of a n-link serial rigid robot manipulator, considering viscous friction, can be expressed as [39],[40]:

$$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+F_v\dot{q}+g\left(q\right)=\tau ,$$

(1)

where q ∊ ℝ ⁿ is the vector of joint positions, $\dot{q}∊{ℝ}^n$ is the vector of joint velocities, $\dot{q}∊{ℝ}^n$ is the vector of joint accelerations, $M\left(q\right)∊{ℝ}^{n\times n}$ is the symmetric positive definite inertia matrix, $C(q,\dot{q})∊{ℝ}^{n\times n}$ is the matrix of centripetal and Coriolis torques, $F_V∊{ℝ}^{n\times n}$ is the diagonal positive definite matrix of viscous friction coefficients, $\tau ∊{ℝ}^n$ is the vector of applied torques, and $g\left(q\right)∊{ℝ}^n$ is the vector of gravitational torques, obtained as the gradient of the robot potential energy U(q), i.e.

$$g\left(q\right)=\frac{\partial U\left(q\right)}{\partial q}.$$

We assume that the links are joined together with revolute joints.

Some important properties of dynamics (1) are the following [39], [40]:

Property 1. The matrix $C(q,\dot{q})$ and the time derivative of the inertia matrix $\dot{M}\left(q\right)$ satisfy [39],[41]:

$${\dot{q}}^T\left[\frac{1}{2}\dot{M}\left(q\right)-C\left(q,\dot{q}\right)\right]=0,$$

for all $q,\dot{q}∊{ℝ}^n$ , and

$$M\left(q\right)=C\left(q,\dot{q}\right)+{C(q,\dot{q})}^T.$$

Property 2. There exists a positive constant k _c such that for all $x,y,z∊{ℝ}^n$ :

$$‖C(x,y)z‖\le k_c‖y‖‖z‖.$$

Property 3. The gravitational torque vector g(q) is bounded for all $q∊{ℝ}^n$ [42]. This means that there exist constants ${\gamma }_i>0$ such that

$$\left|g_i\left(q\right)\right|\le {\gamma }_i,$$

for all $q∊{ℝ}^n$ , where $g_i\left(q\right)$ stands for the i-th element of vector g(q). Equivalently, there exists a positive constant $k^{\prime }>0$ such that

$$‖g\left(q\right)‖\le k^{\mathrm{\prime }}$$

for all $q∊{ℝ}^n$ .

Property 4. There exists a positive constant k _g such that

$$‖g\left(x\right)-g\left(y\right)‖\le k_g‖x-y‖$$

for all $x,y∊{ℝ}^n$ .

2.2 Problem formulation

Consider the robot dynamics (1). Assume that each actuator is capable of supplying a maximum torque ${\tau }_i^{\mathrm{m}\mathrm{a}\mathrm{x}}$ so that:

$$\left|{\tau }_i\right|\le {\tau }_i^{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(2)

for all $i=\mathrm{1,2},\dots ,n,$ , where τ _i stands for the i-th element of applied torques vector τ. We also assume that the maximum torque of each actuator satisfies

$${\tau }_i^{\mathrm{m}\mathrm{a}\mathrm{x}}\ge {\gamma }_i,$$

where γ _i was defined in Property 3. This assumption means that the actuators are capable of holding the robot at rest at any desired joint positions vector q _d .

The control problem is to design a controller to compute the torque vector τ applied to the joints, satisfying (2), such that the joint positions vector q tends asymptotically toward the desired joint positions vector q _d .

2.3 Saturation functions

Before presenting the main contribution of this paper, we present some defintions of the saturation functions that we use throughout our work.

Definition 1. [9] $\mathcal{F}(m,∊,x)$ with $1\ge m>0$ , $∊>0$ , and $x∊{ℝ}^n$ denotes the set of all continously differentiable increasing functions

$$\mathrm{s}\mathrm{a}\mathrm{t}\left(x\right)=\left[\begin{array}{c}\mathrm{s}\mathrm{a}\mathrm{t}\left(x_1\right)\\ \mathrm{s}\mathrm{a}\mathrm{t}\left(x_2\right)\\ \begin{array}{c}⋮\\ \mathrm{s}\mathrm{a}\mathrm{t}\left(x_n\right)\end{array}\end{array}\right]$$

such that

$\left|x\right|\ge \left|\mathrm{s}\mathrm{a}\mathrm{t}\left(x\right)\right|\ge m\left|x\right|$ , for all $x∊ℝ:\left|x\right|<∊,$ ,

$∊\ge \left|\mathrm{s}\mathrm{a}\mathrm{t}\left(x\right)\right|\ge m∊,$ , for all $x∊ℝ:\left|x\right|\ge ∊,$

$1\ge \frac{d\mathrm{s}\mathrm{a}\mathrm{t}\left(x\right)}{dx}\ge 0,$ , for all $x∊ℝ.$

Definition 2. The hard saturation function $SAT(x;\delta )∊{ℝ}^n$ is defined by

$$\begin{array}{l}\text{SAT}\left(x;\delta \right)=\left[\begin{array}{c}\text{SAT}\left(x_1\right)\\ \text{SAT}\left(x_2\right)\\ \begin{array}{c}⋮\\ \text{SAT}\left(x_n\right)\end{array}\end{array}\right],\hfill \\ x=\left[\begin{array}{c}{x}_1\\ x_2\\ \begin{array}{c}⋮\\ x_n\end{array}\end{array}\right],\hfill \\ \delta =\left[\begin{array}{c}{\delta }_1\\ {\delta }_2\\ \begin{array}{c}⋮\\ {\delta }_n\end{array}\end{array}\right],\hfill \end{array}$$

where δ _i is the i-th saturation limit, and

$$\text{SAT}\left(x_i;{\delta }_i\right)=\{\begin{array}{c}{x}_i \text{if} \left|x_i\right|\le {\delta }_i,\\ {\delta }_i \text{if} x_i>{\delta }_i,\\ -{\delta }_i \text{if} x_i<-{\delta }_i,\end{array}$$

for i = 1,2,…,n.

Some properties of the saturation functions defined above are:

Property 5. [9] From Definition 1, it can be proved that:

$$‖\mathrm{s}\mathrm{a}\mathrm{t}\left(x\right)‖\le \left\{\begin{array}{c}‖x‖,\\ ∊,\end{array}\right.$$

for all $x∊{ℝ}^n$ , and

$$‖\mathrm{s}\mathrm{a}\mathrm{t}\left(\mathrm{\Gamma }x\right)‖\le \left\{\begin{array}{c}‖\mathrm{\Gamma }x‖,\\ ∊\sqrt{n},\end{array}\right.$$

for all $x∊{ℝ}^n$ , where Γ is a diagonal positive definite matrix.

Property 6. [9] From Definition 1, we have that:

$$‖\dot{\mathrm{s}\mathrm{a}\mathrm{t}}\left(x\right)‖\le ‖\dot{x}‖,\mathrm{ }\forall x∊{ℝ}^n,$$

and

$$‖\dot{\mathrm{s}\mathrm{a}\mathrm{t}}\left(\mathrm{\Gamma }x\right)‖\le ‖\mathrm{\Gamma }\dot{x}‖,\mathrm{ }\forall x∊{ℝ}^n$$

where Γ is a diagonal positive definite matrix, and the notation $\dot{y}=\frac{d}{dt}y$ has been used.

Property 7. [25] From Definition 1 and 2 it is easy to see that, for all δ _i  ε with i = 1,2,…,n, we have:

$$‖x‖\ge ‖\mathrm{S}\mathrm{A}\mathrm{T}(x;\delta )‖\ge ‖\mathrm{s}\mathrm{a}\mathrm{t}\left(x\right)‖,\mathrm{ }\forall x∊{ℝ}^n,$$

and

$$‖\mathrm{\Gamma }x‖\ge ‖\mathrm{S}\mathrm{A}\mathrm{T}(\mathrm{\Gamma }x;\delta )‖\ge ‖\mathrm{s}\mathrm{a}\mathrm{t}\left(\mathrm{\Gamma }x\right)‖,\mathrm{ }\forall x∊{ℝ}^n,$$

where Γ is a diagonal positive definite matrix.

Property 8. [25] The hard saturation function satisfies, for all k>0:

$$y\left[\mathrm{S}\mathrm{A}\mathrm{T}\left(x+ky;\delta \right)-\mathrm{S}\mathrm{A}\mathrm{T}\left(x;\delta \right)\right]\ge 0,\forall x,y∊ℝ.$$

Property 9. [43] The hard saturation function satisfies, for all x,y ∊ ℝ ⁿ :

$$‖\mathrm{S}\mathrm{A}\mathrm{T}\left(x;\delta \right)-\mathrm{S}\mathrm{A}\mathrm{T}(y;\delta )‖\le ‖x-y‖.$$

Property 10. [25] The integral of a hard saturation function $\mathrm{{\rm I}}\left(x_i\right)-\underset{x_i}{\min } \mathrm{{\rm I}}\left(x_i\right)$ is a positive definite and radially unbounded function with a unique and global minimum at x _i = b/k, with |b|<δ _i , where

$$\mathrm{{\rm I}}\left(x_i\right)={\int }_b^{x_i}\left[\mathrm{S}\mathrm{A}\mathrm{T}\right(k{\sigma }_i;{\delta }_i)-b]d{\sigma }_i.$$

3. Main contribution

In this section we present the main contribution of the paper: a nonlinear PID controller with bounded torques that yields global asymptotical stability, providing a solution to the global set-point control of industrial robot manipulators, that uses the classical PID, but now considering the saturation effects in every control stage and torque constraints of the actuators.

Fig. 1 shows the general scheme of the proposed controller. The controller is formed by two control loops: an outer joint position proportional (P) loop and an inner joint velocity proportional—integral (PI) loop. The scheme considers saturation effects of the commands suppplied by the computer, as well as torque limitations of the actuators and servo-drivers.

3.1 Control scheme

The classical PID controller, considering saturation effects as shown in Fig. 1, may equivalently be expressed as:

$$\tau =\mathrm{S}\mathrm{A}\mathrm{T}\left[K_p^{\mathrm{*}}\mathrm{S}\mathrm{A}\mathrm{T}\left(K_{pc}\stackrel{\sim }{q};{\tau }_p\right)-K_v^{\mathrm{*}}\dot{q}+\mathrm{S}\mathrm{A}\mathrm{T}\left(K_i\omega ;{\tau }_I\right);{\tau }_{pid}\right].$$

(3)

Omitting the saturation limits in (3) we have:

$$\tau =\mathrm{S}\mathrm{A}\mathrm{T}\left[K_p^{\mathrm{*}}\mathrm{S}\mathrm{A}\mathrm{T}\left(K_{pc}\stackrel{\sim }{q}\right)-K_v^{\mathrm{*}}\dot{q}+\mathrm{S}\mathrm{A}\mathrm{T}\left(K_i\omega \right)\right],$$

with

$$\begin{array}{l}\omega =\underset{0}{\overset{t}{{\displaystyle \int }}}\left[\alpha \text{sat}\left(K_{pc}\tilde{q}\left(r\right)\right)-\dot{q}\left(r\right)\right]dr,\hfill \\ K_p^{\text{*}}=\alpha K_p,\hfill \\ K_v^{\text{*}}=K_p,\hfill \end{array}$$

where τ _p , τ _I and τ _pid are the saturation limits vectors such that from the limitations of the real actuators, τ _{pid i}  τ _i ^max for i = 1,2,…,n. K _p , K _pc and K _i are diagonal positive definite matrices whose elements are k _{p i} , k _{pc i} and k _{i i} respectively for $i=\mathrm{1,2},\dots ,n$ , $\stackrel{\sim }{q}=q_d-q$ is the joint position errors vector, the saturation functions sat and SAT were established in Definition 1 and 2, respectively, and α is a small positive constant suitably selected.

The closed-loop equation is obtained by substituting control law (3) in robot dynamics (1):

$$\begin{array}{l}\begin{array}{cc}\frac{d}{dt}\tilde{q}=-\dot{q}& \left(4\mathrm{a}\right)\end{array}\hfill \\ \frac{d}{dt}\dot{q}=M{\left(q\right)}^{-1}[\text{SAT}[K_p^{*}\text{SAT}\left(K_{pc}\widetilde{q;} {\tau }_p\right)-K_v^{*}\dot{q}\hfill \\ +\text{SAT}\left(K_i\omega ; {\tau }_I\right); {\tau }_{pid}]-F_v\dot{q}-C\left(q,\dot{q}\right)\hfill \\ -g(q)]\left(4\mathrm{b}\right)\hfill \\ \begin{array}{cc}\frac{d}{dt}\omega =\alpha \text{sat}\left(K_{pc}\tilde{q}\right)-\dot{q}.& \left(4\mathrm{c}\right)\end{array}\hfill \end{array}$$

(4)

We now present some assumptions that are required in order to prove the main result.

Assumption 1. Saturation limits are chosen so that they satisfy:

$${{\tau }_{pid}}_i>k_{p_i}^{\mathrm{*}}{\tau }_{p_i}+{\tau }_{I_i}$$

(5)

$$\begin{array}{l}{\tau }_{I_i}>{\gamma }_i\hfill \\ {\tau }_{pi{d}_i}<{\tau }_i^{\text{max}}\hfill \end{array}$$

(6)

for all i = 1,2,…, n, with γ _i defined in Property 3. From (5) and (6) it can also be observed that τ _{pid i} >γ _i .

Assumption 2. Matrix K^* _p is chosen such that it satisfies:

$${\lambda }_{min}\left\{K_p\right\}>k_{h2},$$

where

$$k_{h2}=\frac{2k^{\mathrm{\prime }}}{\mathrm{s}\mathrm{a}\mathrm{t}\left(\frac{2k^{\mathrm{\prime }}{\lambda }_{min}\left\{K_{pc}\right\}}{k_g}\right)}.$$

Assumption 3. Positive constant α is chosen small enough so that it satisfies

$$\alpha <\underset{}{\min } \left\{\sqrt{\frac{2\beta {\lambda }_{min}\left\{M\right\}}{{\lambda }_{max}^2\left\{M\right\}}},\frac{4\left[{\lambda }_{min}\left\{K_p\right\}-k_{h2}\right]{\lambda }_{min}\left\{F_v\right\}}{4\left[{\lambda }_{min}\left\{K_p\right\}-k_{h2}\right]a+{\lambda }_{min}^2\left\{K_v^{\mathrm{*}}\right\}}\right\},$$

where

$$a=∊k_c\sqrt{n}+{\lambda }_{max}\left\{K_{pc}\right\}{\lambda }_{max}\left\{M\right\}.\mathrm{ }$$

(7)

Notice that the equilibrium points of the closed-loop system (4) are given by:

$$\begin{array}{l}\dot{q}=0,\hfill \\ \tilde{q}=0,\hfill \\ SAT\left(SAT\left(K_i\omega ;{\tau }_I\right);{\tau }_{pid}\right)=g\left(q_d\right).\hfill \end{array}$$

(8)

Under Assumption 1, (8) is equivalent to:

$$K_i\omega =g\left(q_d\right).$$

Therefore, the only equilibrium point of (4) is given by:

$$\left[\begin{array}{c}\stackrel{\sim }{q}\\ \dot{q}\\ \omega \end{array}\right]=\left[\begin{array}{c}0\\ 0\\ K_i^{-1}g\left(q_d\right)\end{array}\right].$$

(9)

We are now ready to present our main result.

Proposition 1. Controller (3) in closed loop with robot dynamics (1), under Assumptions 1, 2 and 3, yields global asymptotical stability of equilibrium point (9), globally achieving the control objective, and applied torques remain bounded according to (2).

Proof. Proof of Proposition 1 is carried out through Lyapunov stability theory. We propose the Lyapunov function candidate:

$$V\left(\stackrel{\sim }{q},\dot{q},\omega \right)=V_1\left(\stackrel{\sim }{q},\dot{q}\right)+V_2\left(\stackrel{\sim }{q}\right)+V_3\left(\omega \right)-{\mathrm{{\rm I}}}_{\mathrm{m}\mathrm{i}\mathrm{n}},$$

Where

$$\begin{array}{l}{V}_1\left(\tilde{q},\dot{q}\right)=\frac{1}{2}{\dot{q}}^{T}M\left(q\right)\dot{q}-\alpha \text{sat}{(K_{pc}\tilde{q})}^{T}M\left(q\right)\dot{q}+h\left(\tilde{q}\right),\hfill \\ V_2\left(\tilde{q}\right)={\displaystyle \sum }_{i=1}^n\underset{0}{\overset{{\tilde{q}}_i}{{\displaystyle \int }}}{f}_{v_i}\text{sat}\left(k_{p{c}_i}{r}_i\right)d{r}_i,\hfill \\ V_3\left(\omega \right)={\displaystyle \sum }_{i=1}^n\underset{g_i\left(q_d\right)}{\overset{{\omega }_i}{{\displaystyle \int }}}\left[\text{SAT}\left(k_{i_i}{r}_i;{\tau }_{I_i}\right)-g_i\left(q_d\right)\right]d{r}_i,\hfill \\ {\text{Ι}}_{\text{min}}=-{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\frac{g_i{(q_d)}^2}{2}\left[\frac{{\left(k_{I_i}-1\right)}^2}{k_{I_i}}\right],\hfill \\ \begin{array}{l}h\left(\tilde{q}\right)={\displaystyle \sum }_{i=1}^n\underset{0}{\overset{{\tilde{q}}_i}{{\displaystyle \int }}}\text{SAT}\left(k_{p{c}_i}{r}_i;{\tau }_{p_i}\right)d{r}_i+U\left(q\right)-U\left(q_d\right)\\ \begin{array}{c}+g{(q_d)}^T\tilde{q}.\end{array}\end{array}\hfill \end{array}$$

From Property 10 it is straightforward to see that $V_2\left(\stackrel{\sim }{q}\right)$ and V₃(ω) are globally positive definite and radially unbounded functions with respect to $\stackrel{\sim }{q}$ and ω, respectively. Positive definiteness of function $V_1\left(\stackrel{\sim }{q},\dot{q}\right)$ may be proved by lower-bounding each of its terms:

$$\begin{array}{l}{V}_1\left(\tilde{q},\dot{q}\right)\ge \frac{1}{2}{\lambda }_{\text{min}}\left\{M\right\}{\Vert \dot{q}\Vert }^2-\alpha {\lambda }_{\text{max}}\left\{M\right\}\Vert \text{sat}\left(K_{pc}\tilde{q}\right)\Vert \dot{q}\Vert \hfill \\ +\beta {\Vert \text{sat}\left(K_{pc}\tilde{q}\right)\Vert }^2\hfill \\ =-\left[\begin{array}{cc}\Vert \text{sat}\left(K_{pc}\tilde{q}\right)\Vert & \Vert \dot{q}\Vert \end{array}\right]P\left[\begin{array}{c}\Vert \text{sat}\left(K_{pc}\tilde{q}\right)\Vert \\ \Vert \dot{q}\Vert \end{array}\right],\hfill \end{array}$$

where

$$P=\left[\begin{array}{cc}\beta & -\alpha \frac{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{M\right\}}{2}\\ -\alpha \frac{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{M\right\}}{2}& \frac{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{M\right\}}{2}\end{array}\right],$$

and β is a positive constant such that $h(\stackrel{\sim }{q},\dot{q})\ge {\beta ‖sat\left(K_{pc}\stackrel{\sim }{q}\right)‖}^2$ provided that Assumption 2 is satisfied. Under Assumption 3, P is a positive definite matrix. Therefore, we can conclude positive definiteness and radial unboundedness of Lyapunov function $V\left(\stackrel{\sim }{q},\dot{q}\right).$ .

Notice that, under Assumption 1

$$\left|k_{p_i}^{\text{*}}\text{SAT}\left(k_{p{c}_i}{\tilde{q}}_i;{\tau }_{p_i}\right)+\text{SAT}\left(k_{i_i}{\omega }_i;{\tau }_{I_i}\right)\right|\le k_{p_i}^{\text{*}}{\tau }_{p_i}+{\tau }_{I_i}<{\tau }_{pi{d}_i}.$$

Therefore, from Definition 2:

$$\begin{array}{l}\text{SAT}\left[k_{p_i}^{\text{*}}\text{SAT}\left(k_{p{c}_i}{\tilde{q}}_i;{\tau }_{p_i}\right)+\text{SAT}\left(k_{i_i}{\omega }_i;{\tau }_{I_i}\right);{\tau }_{pi{d}_i}\right]\hfill \\ =k_{p_i}^{\text{*}}\text{SAT}\left(k_{p{c}_i}{\tilde{q}}_i;{\tau }_{p_i}\right)+\text{SAT}\left(k_{i_i}{\omega }_i;{\tau }_{I_i}\right),\hfill \end{array}$$

or, equivalently

$$\begin{array}{l}\text{SAT}\left[K_p^{\text{*}}SAT\left(K_{pc}\tilde{q};{\tau }_p\right)+\text{SAT}\left(K_i{\omega }_i;{\tau }_I\right);{\tau }_{pid}\right]\hfill \\ =K_p^{\text{*}}\text{SAT}\left(K_{pc}\tilde{q};{\tau }_p\right)+\text{SAT}\left(K_i\omega ;{\tau }_I\right).\hfill \end{array}$$

(10)

Then, considering (10), and adding and substracting the term $\alpha \mathrm{s}\mathrm{a}\mathrm{t}\left(K_{pc}\stackrel{\sim }{q}{)}^T{K}_p^{*}SAT\right(K_{pc}\stackrel{\sim }{q};{\tau }_p)$ , we can write the time derivative of $V(\stackrel{\sim }{q},\dot{q},\omega )$ along the trajectories of (4) as:

$$\begin{array}{l}\begin{array}{c}\dot{V}\left(\tilde{q},\dot{q},\omega \right)=-{\dot{q}}^T{F}_v\dot{q}-\alpha \text{sat}{(K_{pc}\tilde{q})}^T\left[g\left(q_d\right)-g\left(q\right)\right]\end{array}\hfill \\ -\alpha \text{sat}{(K_{pc}\tilde{q})}^{T}C{(q,\dot{q})}^T\dot{q}\hfill \\ -\alpha \dot{\text{sat}}{(K_{pc}\tilde{q})}^{T}M\left(q\right)\dot{q}\hfill \\ +{\left[\dot{q}-\alpha \text{sat}\left(K_{pc}\tilde{q}\right)\right]}^{T}f\left(\tilde{q},\dot{q},\omega \right)\hfill \\ -\alpha \text{sat}{(K_{pc}\tilde{q})}^T{K}_p^{\text{*}}\alpha \text{SAT}\left(K_{pc}\tilde{q};{\tau }_p\right),\hfill \end{array}$$

(11)

where

$$\begin{array}{l}f\left(\tilde{q},\dot{q},\omega \right)=\text{ SAT}[K_p^{\text{*}}\text{SAT}\left(K_{pc}\tilde{q};{\tau }_p\right)-K_v^{\text{*}}\dot{q}\hfill \\ +\text{SAT}\left(K_i{\omega }_i;{\tau }_I\right);{\tau }_{pid}]\hfill \\ -\text{SAT}[K_p^{\text{*}}\text{SAT}\left(K_{pc}\tilde{q};{\tau }_p\right)\hfill \\ +\text{SAT}\left(K_i{\omega }_i;{\tau }_I\right);{\tau }_{pid}].\hfill \end{array}$$

Negative semidefiniteness of $\dot{V}\left(\tilde{q},\dot{q},\omega \right)$ may be proved by upper-bounding each of the terms in (11):

$-{\dot{q}}^T{F}_v\dot{q}\le -{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{F_v\right\}{‖\dot{q}‖}^2$

$-\alpha \mathrm{s}\mathrm{a}\mathrm{t}(K_{pc}\stackrel{\sim }{q}{)}^T\left[g\left(q_d\right)-g\left(q\right)\right]\le \alpha k_{h2}{‖\mathrm{s}\mathrm{a}\mathrm{t}\left(K_{pc}\stackrel{\sim }{q}\right)‖}^2,$

$-\alpha \mathrm{s}\mathrm{a}\mathrm{t}\left(K_{pc}\stackrel{\sim }{q}{)}^{T}C\right(q,\dot{q}{)}^T\dot{q}\le \alpha ∊k_{h2}\sqrt{n}{‖\mathrm{s}\mathrm{a}\mathrm{t}\left(K_{pc}\stackrel{\sim }{q}\right)‖}^2,$ ,

${\dot{q}}^{T}f\left(\stackrel{\sim }{q},\dot{q},\omega \right)\le 0$ ,

$-\alpha \mathrm{s}\mathrm{a}\mathrm{t}(K_{pc}\stackrel{\sim }{q}{)}^{T}f\left(\stackrel{\sim }{q},\dot{q},\omega \right)\le \alpha {\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{K_v^{\mathrm{*}}\right\}‖\mathrm{s}\mathrm{a}\mathrm{t}\left(K_{pc}\stackrel{\sim }{q}\right)‖‖\dot{q}‖$ ,

$-\alpha \dot{\text{sat}}{(K_{pc}\tilde{q})}^{T}M(q)\stackrel{}{\stackrel{.}{q}\le \alpha {\lambda }_{\text{max}}}\left\{K_{pc}\right\}{\lambda }_{\text{max}}\left\{M\right\}\Vert q{\Vert }^2,$

$-\alpha \text{sat}{(K_{pc}\tilde{q})}^T{K}_p^{\text{*}}\alpha \mathrm{SAT}(K_{pc}\tilde{q};{\tau }_p)\le -\alpha {\lambda }_{\text{min}}\left\{K_p^{\text{*}}\right\}{\Vert sat\left(K_{pc}\tilde{q}\right)\Vert }^2,$

where Properties 2-9 were used. Therefore, $\dot{V}\left(\stackrel{\sim }{q},\dot{q},\omega \right)$ may be upper-bounded by

$$\dot{V}\left(\stackrel{\sim }{q},\dot{q},\omega \right)\le -\alpha \left[\begin{array}{cc}‖\mathrm{s}\mathrm{a}\mathrm{t}\left(K_{pc}\stackrel{\sim }{q}\right)‖& ‖\dot{q}‖\end{array}\right]Q\left[\begin{array}{c}‖\mathrm{s}\mathrm{a}\mathrm{t}\left(K_{pc}\stackrel{\sim }{q}\right)‖\\ ‖\dot{q}‖\end{array}\right],$$

(12)

where

$$Q=\left[\begin{array}{cc}{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{K_p\right\}-k_{h2}& -\frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{K_v\right\}}{2}\\ -\frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{K_v\right\}}{2}& \frac{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left\{F_v\right\}}{\alpha }-a\end{array}\right],$$

and a has been defined in (7). Under Assumptions 2 and 3, Q is a positive definite matrix. Negative semidefiniteness of $\dot{V}\left(\stackrel{\sim }{q},\dot{q},\omega \right)$ is concluded. Noting from (12) that V = 0 if and only if $\tilde{q}=\dot{q}=0$ , it is straightforward to prove global asymptotical stability of equilibrium point (9) via Lasalle's Invariance Principle [43], which completes the proof.

4. Experimental results

Experimental tests were carried out to show the performance of the proposed nonlinear PID with bounded torques. The control scheme was tested in a prototype of two-revolute joint robot manipulator used in [44], located at Instituto Tecnologico de la Laguna (see Figure 2).

Figure 2.

Prototype robot.

For comparison purposes, tests were also carried out with the classical PID controller, given by:

$$\tau =K_p\stackrel{\sim }{q}-K_v\dot{q}+K_i\omega ,$$

(13)

where

$$\dot{\omega }=\stackrel{\sim }{q},$$

and K _p , K _v and K _i are n × n positive definite diagonal matrices.

The desired joint positions were chosen as q _{d 1} = 45 [deg] and q _{d 2} = 45 [deg]. Controller parameters are shown in Table 1.

Table 1.

Controller parameters.

Saturated PID
Controller parameters
k p 1 = 300[Nm/rad]	k p 2 = 120 [Nm/rad]
k v 1 = 35 [Nm sec/rad]	k v 2 = 2 [Nm sec/rad]
k pc 1 = 1000	k pc 2 = 300
k i 1 = 1 [Nm/(sec rad)]	k i 2 = 0.1 [Nm/(sec rad)]
Additional parameters
τ pid 1 = 145.0 [Nm]	τ pid 2 = 14.5 [Nm]
τ p 1 = 50 [rad]	τ p 2 = 10 [rad]
τ I i = 42.5 [Nm]	τ I 2 = 2.0 [Nm]
τmax1 = 150.0 [Nm]	τ2max = 15.0 [Nm]
α = 8.0 [sec]
Classical PID
Controller parameters
k p 1 = 191[Nm/rad]	k p 2 = 19.1 [Nm/rad]
k v 1 = 20 [Nm sec/rad]	k v 2 = 2 [Nm sec/rad]
k i 1 = 150 [Nm/(sec rad)]	k i 2 = 10 [Nm/(sec rad)]

Figs. 3 and 4 show the time response of joint 1 and 2 position errors, respectively, for both, the proposed saturated PID in equation (3) and the classical PID in (13); it can be observed that position errors in the saturated PID case converge faster to zero (1.5 sec. for joint 1 and 0.7 sec. for joint 2). In contrast, the position errors of the classical PID controller arrive faster close to zero, but hold nonzero position errors in the first 5 seconds. The position errors go to zero in a time greater than 5 seconds.

Figure 3.

Joint 1 position error.

Figure 4.

Joint 2 position error.

Figure 5.

Joint 1 applied torque.

Figure 6.

Joint 2 applied torque.

Figs. 5 and 6 show the applied torque to joint 1 and 2, respectively, for both controllers; it can be observed that applied torques remain within the maximum torques supplied by the actuators. The reason why the position errors with the new saturated PID controller (3) go to zero faster than with the classical PID (13) is the feature of using larger proportional gains. In the classical PID controller (13) it was not possible to increase the proportional gains because the maximum torque that can be provided by the actuators would be exceeded.

5. Conclusion

In this paper we analized the stability of the classical PID controller taking into account the natural saturation effects of the commands supplied by the computer and the torque applied by the servo-drivers. By considering these saturation effects the classical PID converts into a new nonlinear PID controller with bounded torques. Global asymptotic stability of the closed-loop system was proved via Lyapunov stability theory. Experimental results were presented in order to observe the performance of the proposed controller.