PID Control of Planar Nonlinear Uncertain Systems in the Presence of Actuator Saturation

Xujun Lyu and Zongli Lin,

Abstract—This paper investigates PID control design for a class of planar nonlinear uncertain systems in the presence of actuator saturation. Based on the bounds on the growth rates of the nonlinear uncertain function in the system model, the system is placed in a linear differential inclusion. Each vertex system of the linear differential inclusion is a linear system subject to actuator saturation. By placing the saturated PID control into a convex hull formed by the PID controller and an auxiliary linear feedback law, we establish conditions under which an ellipsoid is contractively invariant and hence is an estimate of the domain of attraction of the equilibrium point of the closed-loop system. The equilibrium point corresponds to the desired set point for the system output. Thus, the location of the equilibrium point and the size of the domain of attraction determine, respectively, the set point that the output can achieve and the range of initial conditions from which this set point can be reached. Based on these conditions, the feasible set points can be determined and the design of the PID control law that stabilizes the nonlinear uncertain system at a feasible set point with a large domain of attraction can then be formulated and solved as a constrained optimization problem with constraints in the form of linear matrix inequalities (LMIs). Application of the proposed design to a magnetic suspension system illustrates the design process and the performance of the resulting PID control law.

I. INTRODUCTION

PID control has been reported to be the most widely adopted controller in engineering practice [1]. Analysis and design of PID control systems as well as the relationship of PID control with other control design methods continue to be an active research area (see, [2]–[10], for a small sample of the literature). Recently, the design of PID controllers for a class of second order nonlinear uncertain systems was investigated in [11]. Based on the bounds of the growth rates of the nonlinear uncertain function, a 3-dimensional manifold was constructed within which the three PID control coefficients can be chosen arbitrarily that guarantee global asymptotic stability of the closed-loop system at an equilibrium point corresponding to a desired set point for the output. The work of [11] has since motivated further works(see, for example, [12]–[14]).

Inspired by [11], [15], this paper considers the design of PID control for nonlinear uncertain systems in the presence of actuator saturation. Actuator saturation is ubiquitous in realworld control systems. The integral action in a PID controller causes the control input signal to continue to increase even when the system is operating in the vicinity of its steady state,making the actuator in a PID control system especially prone to saturation. It is well-known that actuator saturation degrades the performance of the closed-loop system and, in a severe situation, may even cause the loss of stability. As a result, there have been continual efforts spent in addressing issues associated with actuator saturation (see, for example,[16]–[19], for a small sample of the literature).

The systems we consider will be the same as in [11].Because of the presence of actuator saturation, we are no longer able to achieve global asymptotic stabilization at an equilibrium point as in [11] or may not even be able to achieve local asymptotic stability at some equilibrium points.The location of the equilibrium point and the size of the domain of attraction determine the set point that the output can achieve and the range of initial conditions from which this set point can be reached. Thus, our design goal is to determine feasible set points for the output and the PID coefficients that will achieve local asymptotic stability at the equilibrium point corresponding to a feasible set point for the output with a large domain of attraction.

To meet our design goal, we will first place the system in a linear differential inclusion based on the growth rates of the nonlinear uncertain function in the system model. Each vertex system of the linear differential inclusion is a linear system subject to actuator saturation. Then, by placing the saturated PID control into a convex hull formed by the PID controller and an auxiliary linear feedback law, we establish conditions under which an ellipsoid is contractively invariant and hence is an estimate of the domain of attraction of the equilibrium point of the closed-loop system. Based on these conditions,determination of feasible set points and design of the PID control law that stabilizes the nonlinear uncertain system at a feasible equilibrium point with a large domain of attraction can then be formulated and solved as an optimization with constraints in the form of linear matrix inequalities (LMIs).The PID control coefficients as obtained from solving this constrained optimization problem are shown to take positive values.

Our design algorithm, which involves only solving an LMI problem, would result in a PID controller that stabilizes the system at a feasible equilibrium point corresponding to a feasible set point for the output with the estimate of the domain of attraction maximized. The resulting PID controller is robust with respect to system uncertainties in the sense that it causes the output of any nonlinear system in the class of nonlinear uncertain systems to track the given constant reference. It is expected that the proposed design can be enhanced to achieve performance beyond tracking, such as fast tracking and disturbance rejection.

The remainder of the paper is organized as follows. In Section II, we formulate the problem to be studied in the paper and recall a technical tool that is needed to solve the problem formulated. In Section III, we present our main results on the solution of the problem. Section IV presents an application of the proposed design algorithm to a magnetic suspension test rig. Simulation results are presented that illustrate the effectiveness of the resulting PID controller. Section V draws conclusions to the paper.

II. PROBLEM FORMULATION AND PRELIMINARIES

Consider a planar nonlinear system of the form

where s at is a saturation function defined as

for some known positive scalar ∆ , andf(x1,x2,t) is a nonlinear function that contains uncertainties of the system.We assume that the functionf, withf(0,0,t)=0 andf(x1,0,t)=f(x1,0,0), is locally Lipschitz inx1andx2uniformly intand piecewise continuous int. We also assume that the growth rates offare bounded, that is,

for some known nonnegative constantsL1andL2.

System (1) is commonly found in practice. Examples of such a system include mechanical systems that obey Newton’s second law of motion, such as the magnetic suspension test rig considered in Section IV.

The control objective is to cause the system outputyto track a given constant referencey∗by using a PID controller of the form

wheree(t)=y(t)−y∗, andKP,KI, andKDare the controller coefficients whose values are to be determined. Because of the presence of actuator saturation, we cannot expect that such tracking will occur for any reference outputy∗and from any initial condition of the closed-loop system. Thus, the design objective is to achieve tracking for a largey∗and from a large set of initial conditions.

To deal with the saturation nonlinearity, we adapt from [8]the convex hull representation of a single input saturated linear feedback law.

Lemma 1:LetK,H∈R1×nand δ ∈R. Let

Then, for anyx∈L(H),

where co stands for convex hull.

III. MAIN RESULTS

The closed-loop system under the PID controller (4) can be written as

which, in the absence of saturation, has a unique equilibrium at

where, by (3),

|f(y∗,0,0)|≤L1|y∗|.

We define the following new state variables:

In these new state variables, the closed-loop system can be written as

We note that the tracking problem for system (5) becomes the stabilization problem for system (6) in the new state variables, and. The tracking ability of system (5),b oth the reference outputy∗the system can track and the initial conditions from which the tracking can be achieved, is reflected in the asymptotic stability and the domain of attraction of system (6). Thus, in what follows, we will develop a design algorithm that achieves asymptotic stabilization of system (6) with a large domain of attraction.

where co denotes a convex hull. Thus, the state equation in (6)belongs to the following linear differential inclusion (LDI):

withA(t)∈co{A1,A2,A3,A3}, belongs to the LDI (7).

We have the following result on the asymptotic stability and the domain of attraction of the origin of the state space of the LDI (7). Note that convergence of the stateto the origin implies asymptotic tracking of the desired reference outputy∗by the system output.

Theorem 1:Consider a system described by the LDI (7). LetP∈R3×3be a positive definite matrix and

Suppose that there exists matrixHsuch that

then, the system is asymptotically stable at the origin with Ω(P) contained in the domain of attraction.

Proof:Consider a Lyapunov function candidate

The directive ofValong the trajectory of the closed-loop system (7) within the level set Ω (P)⊂L(H) is given by

Hence, the closed-loop system is asymptotically stable at the origin with Ω (P) contained in the domain of attraction. ■

Corollary 1:Condition (8) of Theorem 1 implies that all elements ofK=[KPKDKI] are positive.

Proof:SincePis positive definite, Condition (8) fori=1 implies thatA1−bBKis a Hurwitz matrix, that is, the coefficients of its characteristic polynomial

are all positive. Thus, all elements ofK,KP,KD, andKIare positive. Based on the conditions established in Theorem 1, the design of the PID control coefficients

can be formulated as a constrained optimization problem,

We note that, by Corollary 1, Constraint b) in the optimization problem (12) ensures that all elements ofKare positive. Thus the positiveness of three PID control coefficientsKP,KD, andKIare automatically satisfied by any solution of the optimization problem.

In the optimization problem, R, referred to a shape reference set, is a pre-specified set used to measure the size of the ellipsoid Ω (P) against. A larger value of α implies a larger set αR that would fit inside Ω(P), which, in turn, implies a larger Ω(P). Examples of the shape reference sets R include a polyhedron

R=co{r1,r2,...,rl}

r1,r2,...,rl∈R3

for some given vectors , and an ellipsoid

for some given positive definite matrixR∈R3×3.

To solve the optimization problem (12), we will transform all its constraints into linear matrix inequalities. If R is a polyhedron, then Constraint a) is equivalent to [17]

where γ =1/α2andQ=P−1. If R is an ellipsoid, Constraint a)is equivalent to

Constraint b) is equivalent to

whereZ=KQ. Similarly, Constraint c) is equivalent to

whereY=HQ.

Finally, recalling that

we have that Constraint d) is implied by

which is equivalent to

With the above transformations, all constraints in (12) are LIMs in the variablesQ,Z,Y, and γ . Thus, when R is a polyhedron, the optimization problem (12) can be transformed into the following LMI problem:

which can be readily solved numerically.

Similarly, when R is an ellipsoid, the optimization problem(12) can be transformed into the following LMI problem:

Let (γ∗,Q∗,Y∗,Z∗) be the solution of the optimization problem (13) or (14), then, we have

IV. APPLICATION TO A MAGNETIC SUSPENSION SYSTEM

Consider the magnetic suspension test rig shown in Fig. 1.The test rig is composed of a beam sitting on a pivot at its center of mass, two active electromagnets located at each end of the beam and non-contacting displacement sensors. The test rig has two rails, mechanical stops at the ends that limit the range of angular motion of the beam to ±0.013 rad(≈ 0.7448°), to protect the coils from being damaged. A motor with an unbalance attached to the shaft is mounted on a track on top of the beam and is removable.

Fig. 1. The beam balancing test rig.

This test rig has been constructed for testing solutions to control problems [20]. In particular, the removable motor with the unbalance, when mounted at different locations along the beam and operated at different rotating speeds, generates periodic signals with different frequencies and magnitudes and thus emulates an exosystem in an output regulation problem [21].

The operation of the test rig is illustrated in Fig. 2. In the figure,I1andI2are the currents in the coils,T1andT2are the torques generated by the electromagnets,Tmis the constant torque induced by the mass of the motor, andTdis a sinusoidal torque caused by the centripetal force keeping the unbalance on its orbit.

Fig. 2. An illustrative diagram for the beam balancing test rig.

The dynamics of the beam, in the absence of the removable motor, can be modeled by the following differential equation:

where θ is the angle between the beam and the horizontal direction,T1andT2are the torques generated by the two electromagnets with the total net torque provided by the electromagnets given byT=T2−T1,Jbis the moment of inertia,kis the stiffness, andDis the damping due to air and pivot friction.

The torquesT1andT2are determined from the air gap fluxes in terms of the coil currentsI1andI2and the beam angle θ as follows:

whereg0is the maximal angle which is reached when one end of the beam touches a rail andcis constant.

Since an electromagnetic force only attracts and cannot repel, the bidirectional total net torque is generated by a differential current. A conventional way to create the differential current is to introduce a bias currentIB>0 and letI1andI2operate symmetrically aroundIB, i.e.,

I1=IB+I

I2=IB−I

where the perturbation currentI, which is small relative to the biased currentIB, is used as a control input that produces the net torque on the beam.

Based on the above derivation, a nominal mathematical model of the test rig was calculated in [22]. This model, in the form of (1), is given as follows:

wherex1=θ,x2=,u=I,a1=9248 ,a2=−1.6335,b=281.9, and ∆=1 A. Because of the inaccuracy in determining the stiffnesskand dampingDof the system, we assume that that the functionfin (1) satisfies (3) withL1=9500 andL2=2.

Suppose that we would like to stabilize the beam at an angley∗=0.003 rad. We will design a PID controller that, given the actuator capacity limited by a saturation function of saturation level of ∆=1 A, causes stabilization from a large set of the initial conditions. In the mean time, for safety considerations,the beam should not hit the mechanical stops, that is,|θ(t)|<0.013,t≥0.

We will carry out our design by solving the constrained optimization problem (13). For Constraint a), we let

R={r1,r2}

withr1=[1 0 0]Tandr2=[0 1 0]T, to reflect the design objective of controlling the beam from both a large initial angle and a large initial angular velocity.

For Constraints b) and c), we have

andb=281.9.

For Constraint d), we have ∆=1,L1=9500,y∗=0.003,andb=281.9.

To ensure that the beam would not hit the mechanical stops,we impose that |x1|<0.013 rad, which can be met by

whereM=[1 0 0]. As a result, we have the following additional constraint for the optimization problem (13):

For practical consideration, we also limit the angular velocity to |x2|≤0.4 rad/sec, which can be met by

whereN=[0 1 0]. Thus, we have another constraint for the optimization problem,

Solving the optimization problem (13) with the additional Constraints e) and f), we obtain

Shown in Fig. 3 is the projection of Ω(P∗) on theplane. As is clear in the figure, both Constraints e) and f) are respected. That is, for any initial condition inside Ω(P∗), the state trajectory will stay inside and the beam would not hit the mechanical stops that protect the coils from being damaged, and the angular velocity of the beam will remain within the range of ± 0.4 rad/sec.

Fig. 3. The projection of Ω (P∗) on the=(θ −0.003,θ˙) plane.

Fig. 4. The evolution of the output (beam angle) of the nominal system,showing asymptotic tracking of the desired output y ∗=0.003 rad.

Fig. 5. The evolution of the beam angular velocity of the nominal system,showing convergence to zero.

Shown in Figs. 4 and 5 are respectively the evolutions of the system outputy(beam angle) and the statex2(beam angular velocity) from an initial condition within Ω (P∗),

In this simulation, we use the nominal values of the system parameters, that is,

As is clear from the simulation results, the PID controller we have designed achieves our design objectives. We note that our design is not by saturation avoidance. We allow the control input to saturate the actuator for full utilization of the actuator capacity, as shown in Fig. 6.

Fig. 6. The evolution of the saturated control input corresponding to the evolutions in Figs. 4 and 5.

To show the robustness of the design to uncertainties in the system, we simulate again withA=A1. The evolutions of the system outputy(beam angle) and the statex2(beam angular velocity) from an initial condition within Ω (P∗),

are shown in Figs. 7 and 8, respectively. The evolution of the actuator output is shown in Fig. 9. These simulation results clearly show the ability of the system output to track its desired value ofy∗=0.003 rad despite the change in the system dynamics.

Fig. 7. The evolution of the output of a system (beam angle) with A = A1,showing asymptotic tracking of the desired output y ∗=0.003 rad.

Fig. 8. The evolution of the beam angular velocity of a system with A=A1,showing convergence to zero.

Fig. 9. The evolution of the saturated control input corresponding to the evolutions in Figs. 7 and 8.

Fig. 10. The evolution of the beam angle of a system with A=(1−sint)A1+(1+sint)A2, showing asymptotic tracking of the desired output y ∗=0.003 rad.

Fig. 11. The evolution of the beam angular velocity of a system with A=(1−sint)A1+(1+sint)A2, showing convergence to zero.

Fig. 12. The evolution of the saturated input corresponding to the evolutions in Figs. 10 and 11.

Fig. 13. The evolution of the beam angle of the nominal system in the presence of nonlinear damping, showing asymptotic tracking of the desired output y ∗=0.003 rad.

Fig. 14. The evolution of the beam angular velocity of the nominal system in the presence of nonlinear damping, showing convergence to zero.

Fig. 15. The evolution of the saturated input corresponding to the evolutions in Figs. 13 and 14.

To show that the design is also effective when the system parameters are time varying, we simulate with

The evolutions of the beam angle, beam angular velocity and the actuator output of the closed-loop system operating from an initial condition within Ω (P∗),

are shown in Figs. 10–12, respectively. These simulation results again show the ability of the system output to track its desired value ofy∗=0.003 rad even when the system dynamics is time-varying.

Finally, we would like to examine the performance of the closed-loop system in the face of nonlinearity in the open-loop system. In particular, we assume that there is nonlinear damping on the beam that has been neglected in the nominal model, that is, the open-loop system is given by

wherex1=θ,x2=,u=I,a1=9248,a2=−1.6335,a3=0.1,andb=281.9. It can be readily verified that, within the invariant level set Ω (P∗), this nonlinear system still belongs to the same LDI for which the PID control coefficients have been designed, and thus the PID controller is valid. To verify this, we simulate the closed-loop system with an initial condition within Ω (P∗),

The evolutions of the system output (beam angle), the beam angular velocity and the actuator output are shown in Figs. 13–15. Once again, we see asymptotic tracking of the desired reference outputy∗by the system output.

V. CONCLUSIONS

In this paper, we revisited the PID control design for a class of planar nonlinear uncertain systems. Motivated by a recent result on the characterization of stabilizing PID control coefficients for a class of planar nonlinear uncertain systems,we considered PID control design in the presence of actuator saturation for such systems. Based on the growth rates of the nonlinear uncertain function in the system model that defines the class of systems, we formulated a constrained optimization problem that searches for the PID control coefficients that, in the presence of actuator saturation, stabilize any nonlinear system in the class with the domain of attraction maximized.The optimization problem was shown to be solvable as an LMI problem. The proposed design algorithm is tested on a magnetic suspension system, which has been constructed for testing solutions to control problems. Extensive simulation results show the effectiveness of the design algorithm.