Semi-global stabilization with guaranteed regional performance of linear systems subject to actuator saturation

https://doi.org/10.1016/S0167-6911(01)00098-6Get rights and content

Abstract

For a linear system under a given saturated linear feedback, we propose feedback laws that achieve semi-global stabilization on the null controllable region while preserving the performance of the original feedback law in a fixed region. Here by semi-global stabilization on the null controllable region we mean the design of feedback laws that result in a domain of attraction that includes any a priori given compact subset of the null controllable region. Our design guarantees that the region on which the original performance is preserved would not shrink as the domain of attraction is enlarged by appropriately adjusting the feedback laws. Both continuous-time and discrete-time systems will be considered.

Introduction

We revisit the problem of semi-globally stabilizing a linear system on its null controllable region with saturating actuators. The null controllable region, denoted as C, is the set of states that can be steered to the origin of the state space in a finite time using saturating actuators. The problem of semi-global stabilization on the null controllable region is, for any a priori given set X that is in the interior of the null controllable region C, to find a stabilizing feedback law u=FX(x) such that the resulting domain of attraction includes X as a subset.

This problem has been well studied for systems that are so-called asymptotically null controllable with bounded controls (ANCBC).2 In particular, it is established in [6], [7] that, in both continuous-time and discrete-time, a linear ANCBC system is semi-globally asymptotically stabilizable on its null controllable region by saturated linear feedback. We note that in this case, the null controllable region is the entire state space. The key to the possibility of achieving semi-global stabilization on C by linear feedback is that the open loop system is ANCBC. In general saturated linear feedback cannot achieve semi-global stabilization on C if the open loop system is not ANCBC, although there have been many attempts to enlarge the domain of attraction by appropriately choosing the linear feedback gains (see, for example, [3] and the references therein).

Our objective in this paper is to construct nonlinear feedback laws that semi-globally stabilize a linear system (not necessarily ANCBC) subject to actuator saturation. This problem has been addressed before. In particular, it was established in [4], [5] that, in both continuous-time and discrete-time, a linear system with only two exponentially unstable modes can be semi-globally stabilized on its null controllable region by controllers that switch between two linear feedback laws. By defining these two linear feedback laws on an appropriately constructed invariant set, it is guaranteed that switching would occur at most once. In discrete-time, general systems have been considered in [1] and feedback laws were constructed that achieve semi-global stabilization on the null controllable region. More specifically, a sequence of polygons are constructed that approaches the null controllable region as the number of vertices increases. The vertices divide the polygons into cones. The state feedback laws are then constructed based on the controls that drive the vertices of a polygon to the origin according to which cone the state belongs to.

In this paper we will first consider a general linear system subject to actuator saturation,x(k+1)=Ax(k)+Bσ(u(k)),x∈Rn,u∈Rm,where σ is the standard saturation function. With a slight abuse of notation, we use the same symbol to denote both the vector saturation function and the scalar saturation function, i.e., if v∈Rm, then σ(v)=[σ(v1),σ(v2),…,σ(vm)]T and σ(vi)=sgn(vi)min{1,|vi|}. We also assume that a feedback law u=F0(x) has been designed such that the resulting closed-loop system in the absence of the saturation functionx(k+1)=Ax(k)+BF0(x(k))has the desired performance. We need to study the stability and performance of the actual system in the presence of actuator saturation,x(k+1)=Ax(k)+Bσ(F0(x(k))).Let D0 be an invariant set of the closed-loop system and be inside the linear region of the saturation function: {x∈Rn:∥F0(x)∥⩽1}. For example, a linear state feedback law u=F0x could be constructed that places the closed-loop poles at certain desired locations and D0 can be a level set of the form {x∈Rn:xTP0x⩽1}, where P0>0 satisfies(A+BF0)TP0(A+BF0)−P0<0.Suppose that D0 is in the linear region, then it is an invariant set and within D0, the saturation function does not have an effect and hence the desired closed-loop performance is preserved.

The objective of this paper is to construct feedback laws that semi-globally stabilize the system (1) on its null controllable region and in the mean time preserve the desired closed-loop performance in the region D0. The structure of our feedback laws is completely different from that of [1]. Instead of resorting to the cones of the polygons which are not invariant sets, we design our controller by combining a sequence of feedback laws u=Fi(x),i=0,1,…,M, in a way that the union of the invariant sets corresponding to each of the feedback laws is also an invariant set, which is shown to be in the domain of attraction. By appropriately selecting this sequence of feedback laws, the union of the invariant sets can then be made large enough to enclose any subset in the interior of the null controllable region. This idea was made feasible by the use of the lifting technique, which was used in [2] to provide an alternative proof of the results of [7] mentioned earlier. We will also extend the above results to continuous-time systems.

This paper is organized as follows. In Section 2 we propose a method for expanding the domain of attraction by switching between a finite sequence of feedback laws. This switching design is then used in Section 3 to show that the domain of attraction can be enlarged to include any subset in the interior of the null controllable region. Section 4 extends the results of Section 3 to continuous-time systems. An example is given in Section 5 to illustrate our design results. Finally, a brief concluding remark is made in Section 6.

Section snippets

Expansion of the domain of attraction

Let u=Fi(x),i=0,1,…,M, be a finite sequence of stabilizing feedback laws. Among these feedback laws, u=F0(x) can be viewed as the feedback law that was originally designed to guarantee certain desired closed-loop performance in a given region and the remaining feedback laws have been introduced for the purpose of enlarging the domain of attraction while preserving the regional performance of the original feedback law u=F0(x).

For each i=0,1,…,M, let Di be an invariant set inside the domain of

A semi-global stabilization strategy

In this section, we utilize the lifting technique to design a sequence of ellipsoids that cover any prescribed compact subset of the null controllable region. Each ellipsoid is invariant and in the domain of attraction for the lifted closed-loop system under an appropriately chosen linear feedback. This, by Theorem 1, would achieve semi-global stabilization for the lifted system, and hence for the original system.

The null controllable region of (1) at step K, denoted as C(K), is the set of

Continuous-time systems

In this section, we consider the continuous-time counterpart of the system (1)ẋ(t)=Ax(t)+Bσ(u(t)),x∈Rn,u∈Rm.The null controllable region at time T, denoted as C(T), is the set of states that can be steered to the origin in time T by a measurable control input u. The null controllable region, denoted as C, is T⩾0C(T).

Let h>0 be the lifting period. We are now interested in controlling the state of (21) at times kh,k=1,2,…. Denote xh(k)=x(kh) and uh(k,τ)=u(kh+τ). Let Ah=eAh; then the lifted

Example

Consider the system (1) withA=0.8876−0.55550.55551.5542,B=−0.11240.5555.The matrix A is exponentially unstable with a pair of eigenvalues 1.2209±j0.4444. The LQR controller corresponding to the cost function J=∑(x(k)TQx(k)+u(k)TRu(k)), with Q=I,R=1 is u=F0(x)=[−0.2630−2.1501]x. Let D0 be obtained asE(P0),P0=2.1367−0.2761−0.27611.7968,see the ellipsoid enclosed by the solid curve in Fig. 1.

To enlarge the domain of attraction, we take a lifting step of 8 and obtain 16 invariant ellipsoids with

Conclusions

In this paper, we have proposed a control design method for linear systems that are subject to actuator saturation. This design method applies to general (possibly exponentially unstable) systems in either continuous-time or discrete-time. The resulting feedback laws expand the domain of attraction achieved by an a priori designed feedback law to include any bounded set in the interior of the null controllable region, while preserving the desired performance of the original feedback law in a

References (7)

There are more references available in the full text version of this article.

Cited by (0)

1

The work of Tingshu Hu and Zongli Lin was supported in part by the US Office of Naval Research Young Investigator Program under grant N00014-99-1-0670.

View full text