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Nonlinear observers: a circle criterion design and robustness analysis
非线性观测器：圆准则设计和鲁棒性分析

Murat Arcak , Petar Kokotović Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY, 12180-3590, USA
美国纽约州特洛伊市，12180-3590，伦斯勒理工学院电气、计算机和系统工程系 Department of Electrical, and Computer Engineering, University of California, Santa Barbara, CA 93106-9560, USA
加利福尼亚大学圣巴巴拉分校电气与计算机工程系，美国加利福尼亚州 93106-9560

Received 1 August 2000; revised 3 May 2001; received in final form 30 May 2001
2000 年 8 月 1 日收到；2001 年 5 月 3 日修订；2001 年 5 月 30 日最终形式接收

Abstract 摘要

Globally convergent observers are designed for a class of systems with monotonic nonlinearities. The approach is to represent the observer error system as the feedback interconnection of a linear system and a time-varying multivariable sector nonlinearity. Using LMI software, observer gain matrices are computed to satisfy the circle criterion and, hence, to drive the observer error to zero. In output-feedback design, the observer is combined with control laws that ensure input-to-state stability with respect to the observer error. Robustness to unmodeled dynamics is achieved with a small-gain assignment design, as illustrated on a jet engine compressor example. (C) 2001 Elsevier Science Ltd. All rights reserved.
全局收敛观测器被设计用于一类具有单调非线性的系统。方法是将观测器误差系统表示为线性系统和时变多变量扇形非线性的反馈互连。使用 LMI 软件，计算观测器增益矩阵以满足圆环准则，从而将观测器误差驱动为零。在输出反馈设计中，观测器与控制律相结合，确保相对于观测器误差的输入状态稳定性。通过小增益分配设计实现对未建模动态的鲁棒性，如喷气发动机压缩机示例所示。 (C) 2001 Elsevier Science Ltd. 保留所有权利。

Keywords: Nonlinear observer design; Output-feedback control; Axial compressors; Robustness; Unmodeled dynamics
关键词：非线性观测器设计；输出反馈控制；轴向压缩机；鲁棒性；未建模动态

1. Introduction 1. 简介

Progress in nonlinear observers has been slower than in other areas of nonlinear control theory. Early results were obtained under a global Lipschitz restriction which excludes common nonlinearities such as , etc. The results of Thau (1973), Kou, Elliott, and Tarn (1975), Banks (1981) and their recent extensions by Raghavan and Hedrick (1994), and Rajamani (1998) make use of quadratic Lyapunov functions for the observer error system. A more advanced framework was introduced by Tsinias , who pursued an observer analog of control Lyapunov functions and used them for output feedback design.
非线性观测器的进展比非线性控制理论的其他领域慢。早期结果是在排除常见非线性如 等的全局 Lipschitz 限制下获得的。Thau（1973）、Kou、Elliott 和 Tarn（1975）、Banks（1981）及其最近由 Raghavan 和 Hedrick（1994）以及 Rajamani（1998）扩展的结果利用二次 Lyapunov 函数来处理观测器误差系统。Tsinias 引入了更先进的框架，追求控制 Lyapunov 函数的观测器模拟，并将其用于输出反馈设计。
Another common restriction is to require that the nonlinearities appear as functions of the measured output. Systems which admit such a representation have
另一个常见的限制是要求非线性出现为测量输出的函数。具有这种表示的系统包括
been characterized by Krener and Isidori (1983), Bestle and Zeitz (1983), and other authors. The observer design for these systems is linear because the nonlinearity is canceled by an output injection term.
被 Krener 和 Isidori（1983），Bestle 和 Zeitz（1983）以及其他作者所描述。这些系统的观测器设计是线性的，因为非线性性被输出注入项抵消了。
A design which dominates state-dependent nonlinearities by high-gain linear terms has been developed by Esfandiari and Khalil (1992). To avoid the destabilizing effect of the peaking phenomenon analyzed by Sussmann and Kokotovic (1991), this design employs judicious scaling of the observer gain matrix, and saturation of the peaking observer signals before they are fed to the controller. To achieve global convergence of high-gain observers, Gauthier, Hammouri, and Othman (1992) resorted to a global Lipschitz restriction.
Esfandiari 和 Khalil（1992）开发了一种通过高增益线性项主导状态相关非线性的设计。为了避免 Sussmann 和 Kokotovic（1991）分析的尖峰现象的破坏效应，该设计采用了观测器增益矩阵的谨慎缩放，并在将其馈送给控制器之前对尖峰观测器信号进行饱和。为了实现高增益观测器的全局收敛，Gauthier，Hammouri 和 Othman（1992）采用了全局 Lipschitz 限制。
The design presented in this paper removes the global Lipschitz restriction and avoids high-gain. Instead, it makes two restrictions which allow the observer error system to satisfy the well-known multivariable circle criterion. First, a linear matrix inequality (LMI) is to be feasible, which implies a strict positive real (SPR) property for the linear part of the observer error system. The second restriction is that the nonlinearities be nondecreasing functions of linear combinations of unmeasured states. This restriction ensures that the vector time-varying nonlinearity in the observer error system satisfies the sector condition of the circle criterion. A preliminary idea of this global observer design was presented in Arcak and Kokotovic (1999) for systems in strict-feedback form. In the current paper we develop a complete design
本文提出的设计去除了全局 Lipschitz 限制并避免高增益。相反，它制定了两个限制条件，允许观测误差系统满足著名的多变量圆准则。首先，要求线性矩阵不等式（LMI）可行，这意味着观测误差系统的线性部分具有严格正实（SPR）特性。第二个限制是非线性函数应为未测量状态的线性组合的非递减函数。这个限制确保观测误差系统中的向量时变非线性满足圆准则的扇形条件。Arcak 和 Kokotovic（1999）在严格反馈形式系统中提出了这种全局观测器设计的初步想法。在本文中，我们开发了一个完整的设计。
procedure for more general systems and provide a robustness analysis for observer-based feedback design.
为更一般的系统提供程序，并为基于观测器的反馈设计进行鲁棒性分析。
In the proposed design, presented in Section 2, observer matrices satisfying the circle criterion are evaluated by efficient LMI computations. The robustness property against inexact modeling of nonlinearities is established, and the feasibility of a reduced-order variant of the observer is shown to be equivalent to that of the full-order observer. In Section 3, the new observer is incorporated in output-feedback design with control laws that ensure input-to-state stability (ISS) with respect to the observer error. For unmodeled dynamics such as those studied by Praly and Jiang (1993), a small-gain assignment design achieves robust observer-based stabilization. As illustrated by a jet engine compressor example in Section 4, this design broadens the applicability of the new observer to systems which violate feasibility of the exact observer design.
在提出的设计中，第 2 节中提出的观测矩阵通过高效的 LMI 计算来评估满足圆形准则的观测矩阵。针对非线性建模不精确性的鲁棒性特性得到了建立，并且显示了观测器的降阶变体的可行性等同于完全阶观测器的可行性。在第 3 节中，新的观测器被纳入到输出反馈设计中，其中控制律确保了相对于观测器误差的输入-状态稳定性（ISS）。对于像 Praly 和 Jiang（1993）研究的未建模动态，小增益分配设计实现了基于鲁棒观测器的稳定。正如第 4 节中喷气发动机压缩机示例所示，这种设计扩展了新观测器的适用性，适用于违反精确观测器设计可行性的系统。

2. Nonlinear observers: circle criterion design
2. 非线性观测器：圆准则设计

For our observer design, we consider the plant
对于我们的观察器设计，我们考虑植物
,
,
where is the state, is the measured output, is the control input, the pair is detectable, and, and are locally Lipschitz. The statedependent nonlinearity is an -dimensional vector where each entry is a function of a linear combination of the states
其中 是状态， 是测量输出， 是控制输入，对 是可检测的，而 和 是局部 Lipschitz 的。所述依赖状态的非线性 是一个 维向量，其中每个条目是状态的线性组合的函数
.
Our main restriction is that each be nondecreasing, that is, for all , it satisfies
我们的主要限制是每个 都是非递减的，也就是说，对于所有 ，它满足
.
If is continuously differentiable, then for all . If, instead, satisfies , , we can still represent the system as in (1)-(3) by defining a new function which satisfies . When the nonlinearity also depends on and , the nondecreasing property (3) is to hold for each and .
如果 是连续可微的，那么对于所有 都成立 。相反，如果 满足 ， ，我们仍然可以通过定义一个新函数 来表示系统，该函数满足 。当非线性 也取决于 和 时，非递减性质（3）应对每个 和 成立。
For the plant (1), we construct an observer in the form
对于该植物（1），我们以观察者的形式构建
.
Our task is to determine the observer matrices and . At this point, we assume that the solution of (1) does not escape to infinity in finite time. In
我们的任务是确定观察者矩阵 和 。在这一点上，我们假设方程（1）的解 在有限时间内不会趋向无穷远。
Section 3, this assumption will be satisfied by our output feedback control design.
第 3 节，我们的输出反馈控制设计将满足这一假设。
From (1) and (4), the dynamics of the observer error are governed by
从（1）和（4）可以看出，观测误差 的动态由以下方程控制：
,
where 哪里
.
We begin the observer design by representing the observer error system (5) as the feedback interconnection of a linear system and a multivariable sector nonlinearity. To this end, we view as a function of and ; that is, a time-varying nonlinearity in :
我们通过将观测器误差系统（5）表示为线性系统和多变量扇区非线性的反馈互连来开始观测器设计。为此，我们将 视为 和 的函数；也就是说， 中的一个时变非线性：
,
where the time dependence is due to . Substituting (7), we rewrite the observer error system (5) as
时间依赖性是由 引起的。将（7）代入，我们将观测者误差系统（5）重写为
and note from (3) that each component of satisfies
请注意从（3）中， 的每个组件都满足
Thanks to this sector property of each , the product is nonnegative for any diagonal . This means that the feedback path in the observer error system depicted in Fig. 1 is a multivariable sector nonlinearity. Thus, from the circle criterion, asymptotic stability is guaranteed if the linear system with input and output is SPR, that is, if a matrix , a constant , and a diagonal matrix exist such that
由于每个 的部门属性，对于任何对角线 ，产品 都是非负的。这意味着在图 1 中所示的观测误差系统中的反馈路径是多变量部门非线性。因此，根据圆环准则，如果具有输入 和输出 的线性系统是 SPR，即存在矩阵 、常数 和对角矩阵 ，则渐近稳定性是有保证的。
Thus, the observer design for system (1) consists in finding observer matrices and to satisfy (10) with some
因此，系统（1）的观测器设计在于找到观测器矩阵 和 ，使得满足（10）的条件
Fig. 1. Observer error system.
图 1. 观察者误差系统。
, and . Efficient numerical tools are available for this task because (10) is a LMI in , and . It is important to note that the feasibility of (10) is not known a priori, and is to be determined numerically. An analytical test of feasibility has been derived in Arcak (2000).
，和 。 由于（10）是在 ， 和 中的 LMI，因此可以使用高效的数值工具来完成此任务。 需要注意的是，（10）的可行性事先是未知的，需要通过数值方法来确定。 可行性的分析测试已在 Arcak（2000）中推导出。
Robustness issues are critical in every observer design. To analyze the robustness of the above observer against inexact modeling of nonlinearities, we suppose that instead of (1), the plant is
鲁棒性问题在每个观测器设计中都是至关重要的。为了分析上述观测器对非线性建模不精确性的鲁棒性，我们假设植物是代替（1）。
,
where is an unknown bounded disturbance. Then, using the observer (4), we get the observer error system
其中 是一个未知的有界扰动。然后，使用观测器（4），我们得到观测器误差系统
,
where and are as in (6).
其中 和 如(6)中所示。
We now characterize uncertain nonlinearities for which the observer (4) guarantees an ISS property from the disturbance to the observer error .
我们现在对不确定的非线性特性 进行表征，其中观测器（4）从干扰 到观测器误差 保证了 ISS 特性。
Theorem 1. Consider the plant (11) and the observer (4). Suppose exists for all , and that the LMI (10) holds with a matrix , a constant , and a diagonal matrix . If, for each , , there exists a class- function such that
定理 1. 考虑植物（11）和观察者（4）。假设对于所有 都存在 ，并且 LMI（10）满足矩阵 ，常数 和对角矩阵 。如果对于每个 ，存在一个类- 函数 ，使得
,
then the observer error satisfies, for all ,
然后观察者误差 满足，对于所有 ，
,
where , and the ISS-gain from to is
在 ，ISS-增益从 到 是
Proof. We use as an ISS-Lyapunov function, and evaluate its derivative for (12):
证明。我们使用 作为 ISS-Lyapunov 函数，并评估其在（12）中的导数：
Substituting (13), we obtain
用（13）替换，我们得到
from which it follows that
由此可见
so that (14) and (15) result from .
使得（14）和（15）的结果来自 。
The ISS property established by Theorem 1 shows that decreases with the decrease in the magnitude of the disturbance . As vanishes, we recover exponential convergence of the observer error to zero.
由定理 1 建立的 ISS 性质表明， 随着扰动 幅度的减小而减小。当 消失时，我们恢复了观测器误差指数收敛到零的情况。
The dependence of uncertain nonlinearities on is characterized by (13). For example, if is cubic, then is allowed to be linear. In this case, (13) is satisfied because
不确定非线性 对 的依赖性由(13)表征。例如，如果 是立方的，那么 就可以是线性的。在这种情况下，(13)被满足，因为
holds for all due to the identity .
由于恒等式 ，对于所有 都成立。
In applications it may be more convenient to employ a reduced-order observer, which generates estimates only for the unmeasured states. The design of such an observer starts with a preliminary change of coordinates such that the output consists of the first entries of the state vector . In the new coordinates, the system (1) is
在应用中，使用降阶观测器可能更方便，它仅为未测量的状态生成估计值。这种观测器的设计始于对坐标的初步变换，使得输出 由状态向量 的前 个条目组成。在新的坐标系中，系统（1）是
,
,
where the linear terms in are incorporated in and . An estimate of will be obtained via , where is to be designed. From (20), the derivative of is
线性项在 中并入 和 。通过 可以获得 的估计，其中 将被设计。从（20）中， 的导数是
where . In this -subsystem, the output injection matrix has altered the and matrices of the -subsystem (20).
在这个 -子系统中，输出注入矩阵 已经改变了 -子系统的 和 矩阵（20）。
To obtain the estimate 获取估计
,
we employ the observer 我们雇用观察者
From (21) and (23), the dynamics of are governed by
从（21）和（23）可以看出， 的动态受到控制
,
where and . We let , and denote
在 和 。我们让 ，并表示
. Then, the nondecreasing property (3) implies that for all , and derivations similar to those in Section 2 yield the following LMI in and :
。然后，非递减性质（3）意味着对于所有 ， ，并且类似于第 2 节中的推导产生以下 和 中的 LMI：
If this LMI is satisfied with a matrix , a constant , and a diagonal matrix , then it is not difficult to show that, with appropriate modifications, Theorem 1 holds for the observer error . Furthermore, as we prove in Appendix A.1, the reduced-order observer exists whenever the full-order observer exists:
如果这个 LMI 满足一个矩阵 ，一个常数 和一个对角矩阵 ，那么很容易证明，通过适当的修改，定理 1 对观测误差 成立。此外，正如我们在附录 A.1 中证明的那样，当全阶观测器存在时，简化阶观测器也存在：
Theorem 2. Let the constant and the diagonal matrix be given. Then, the following two statements are equivalent:
定理 2. 假设给定常数 和对角矩阵 。那么，以下两个陈述是等价的：
  1. There exist matrices and satisfying the full-order observer LMI (10).
    存在矩阵 和 ，满足全阶观测器 LMI（10）。
  2. There exist matrices and satisfying the reduced-order observer LMI (25).
    存在矩阵 和 ，满足降阶观测器 LMI（25）。
To emphasize the importance of the concept of injecting an output signal in the observer nonlinearity , we provide an example in which the observer cannot be designed with :
为了强调在观察者非线性 中注入输出信号的概念的重要性，我们提供一个例子，其中观察者无法设计为 ：
Example 1. With , and the nondecreasing nonlinearity , the system
示例 1。使用 和非递减非线性 ，系统
is of the form (1) with , . Because the off-diagonal term in the LMI (10) must be zero, the restriction implies that be diagonal. However, because
是以 ， 的形式。由于 LMI（10）中的非对角项必须为零，所以限制 意味着 必须是对角的。然而，因为
it is not difficult to show that cannot be negative definite with a positive definite diagonal .
很容易证明 不能是具有正定对角线 的负定。
With the design is feasible and a solution of the LMI (10) is . The resulting observer is
设计是可行的，LMI（10）的解决方案是 。得到的观察器是
,
.
Likewise, a reduced-order observer for is
同样， 的降阶观测器是
.

3. Output-feedback control
3. 输出反馈控制

The design of a nonlinear output-feedback control law using the certainty equivalence implementation of a full state-feedback law is, in general, impossible as we illustrate on the system (26). This system is stabilizable with full state feedback
通常情况下，使用完全状态反馈法的确定等效实现设计非线性输出反馈控制律是不可能的，正如我们在系统（26）上所说明的那样。这个系统可以通过完全状态反馈来稳定。
.
When only is measured, the certainty equivalence control law (29) with the unmeasured state replaced with its estimate results in the closed-loop system
当仅测量 时，将未测量的状态 替换为其估计值 的确定性等效控制法则（29）导致闭环系统
where is exponentially decaying. As we prove in Appendix A.2, this system exhibits finite escape time for any choice of and .
其中 是指数衰减的。正如我们在附录 A.2 中证明的那样，该系统对于任何 和 的选择都表现出有限的逃逸时间。
Rather than attempting a certainty equivalence design, we view the state observer error as a disturbance acting on the system (1), and design a control law to render the system ISS with respect to the observer error . Such ISS control laws have already been designed for classes of nonlinear systems in Freeman and Kokotović (1996), Krstić, Kanellakopoulos, and Kokotovic (1995), and Marino and Tomei (1995). Here, we pursue a more general output-feedback design, where we achieve robustness against dynamic modeling errors. We consider the plant
与尝试确定等效设计不同，我们将状态观测器误差 视为作用于系统（1）的干扰，并设计一个控制律 ，使系统相对于观测器误差 成为 ISS。这样的 ISS 控制律已经在 Freeman 和 Kokotović（1996）、Krstić、Kanellakopoulos 和 Kokotovic（1995）以及 Marino 和 Tomei（1995）的一些非线性系统类别中设计过。在这里，我们追求更一般的输出反馈设计，以实现对动态建模误差的鲁棒性。我们考虑植物。
,
,
,
where and are as in (13), and the -subsystem (32) represents unmodeled dynamics. This formulation extends the applicability of our observer because, if there is a subsystem to which the observer is not applicable, it can be treated as unmodeled dynamics.
其中 和 如(13)中所示， -子系统(32)代表未建模动态。这种表述扩展了我们观测器的适用性，因为如果有一个子系统观测器不适用，可以将其视为未建模动态。
The unmodeled dynamics (32) are assumed to possess the following input-to-output stability (IOS), and ISS properties:
未建模动态（32）被假定具有以下输入到输出稳定性（IOS）和 ISS 属性：
,
,
where are class- functions, and are class- functions.
其中 是类- 函数， 是类- 函数。
The ISS property of the observer error (14) is now rewritten in the 'max' form of Teel (1996), which is
观察者误差的 ISS 性质（14）现在以 Teel（1996）的“max”形式重新书写，这是
Fig. 2. Closed-loop system with observer feedback.
图 2. 具有观测器反馈的闭环系统。
suitable for the small-gain analysis we will employ:
适用于我们将采用的小收益分析：
.
To prepare for a small-gain design of , we represent the closed-loop system with the block-diagram in Fig. 2. The gains of the unmodeled dynamics block and the observer error block are and , respectively. Let and be the plant gains from and to , respectively. The task for the control law is to render and small enough for the inner loop gain , and the outer loop gain to satisfy, for all ,
为了准备 的小增益设计，我们用图 2 中的块图表示闭环系统。未建模动态块和观测器误差块的增益分别为 和 。让 和 分别是从 到 和 的工厂增益。控制定律 的任务是使内环增益 和外环增益 足够小，以满足所有 。
,
.
Then, global asymptotic stability (GAS) of the closedloop system will be guaranteed as in the nonlinear smallgain theorem of Jiang, Teel, and Praly (1994), and Teel (1996).
然后，封闭环系统的全局渐近稳定性（GAS）将得到保证，就像江、蒂尔和普拉利（1994 年）以及蒂尔（1996 年）的非线性小增益定理中所保证的那样。
Theorem 3. Consider the system (31) and (32), in which and satisfy (13), and the -subsystem satisfies (33) and (34). Suppose that the observer
定理 3. 考虑系统(31)和(32)，其中 和 满足(13)，而 -子系统满足(33)和(34)。假设观察者
is such that the LMI (10) holds with a matrix , a constant , and a diagonal matrix . If the control law guarantees
是这样的，LMI（10）以矩阵 ，常数 和对角矩阵 为基础。如果控制法则 保证
where and satisfy (36) and (37), respectively, then the origin of the closed-loop system (31), (32) and (38) is globally asymptotically stable.
当 和 分别满足(36)和(37)时，闭环系统(31)、(32)和(38)的原点是全局渐近稳定的。

4. Design example 4. 设计示例

An axial compressor model, which has been the starting point for jet engine control studies, is the following single-mode approximation of a PDE model due to Moore and Greitzer (1986),
轴向压缩机模型，这是喷气发动机控制研究的起点，是由 Moore 和 Greitzer（1986 年）提出的 PDE 模型的以下单模近似
,
,
,
where and are the deviations of the mass flow and the pressure rise from their set points, the control input is the flow through the throttle, and, and are positive constants. This model captures the main surge instability between the mass flow and the pressure rise. It also incorporates the nonnegative magnitude of the first stall mode.
其中 和 分别是质量流量和压力升高与其设定值的偏差，控制输入 是节流阀的流量， 和 是正常数。该模型捕捉了质量流量和压力升高之间的主要涡轮不稳定性。它还包括第一阻塞模式的非负幅度 。
A state feedback GAS control law in (Krstić et al., 1995, Section 2.4) was replaced in Krstić, Fontaine, Kokotović, and Paduano (1998) by a design using and . With a high-gain observer, (Isidori, 1999, Section 12.7), and Maggiore and Passino (2000) obtained a semiglobal result using the measurement of alone. With , we will now achieve GAS for (41)-(43). While the exact observer cannot be designed because of the nonlinearities and , the -subsystem (41) and (42) contains the nondecreasing nonlinearity , and is of the form (31) with disturbance . This suggests that we treat the -subsystem (43) as unmodeled dynamics and apply the small-gain design of Section 3.
(Krstić等人，1995 年，第 2.4 节)中的状态反馈 GAS 控制法在 Krstić，Fontaine，Kokotović和 Paduano（1998 年）中被设计为使用 和 。通过高增益观测器（Isidori，1999 年，第 12.7 节）和 Maggiore 和 Passino（2000 年）仅使用 的测量获得了半全局结果。通过 ，我们现在将实现（41）-（43）的 GAS。由于非线性 和 ，无法设计精确的观测器， -子系统（41）和（42）包含非递减非线性 ，并且具有扰动 的形式（31）。这表明我们将 -子系统（43）视为未建模动态，并应用第 3 节的小增益设计。
First, we prove that satisfies the IOS property (33) with as the input. With as an ISS-Lyapunov function, implies , because for all . This means that (33) holds with the linear gain
首先，我们证明 满足 IOS 属性（33），输入为 。以 作为 ISS-Lyapunov 函数， 意味着 ，因为 对于所有 。这意味着（33）以线性增益保持。
,
and, since , the ISS property (34) is also satisfied.
自 以来，ISS 性质（34）也得到满足。
To design the reduced-order observer of Section 2 for the ( -subsystem (41) and (42), we let , and obtain
设计第 2 节的降阶观测器，用于（ -子系统（41）和（42）），我们令 ，并获得
where 哪里
.
The resulting observer for is the scalar equation
的结果观察者是标量方程
.
For its implementation, the LMI (25) is satisfied by selecting such that
对于其实施，通过选择 来满足 LMI（25）
.
To prove the ISS property (35) for the observer error , we employ the ISS-Lyapunov function , and evaluate its derivative for
为了证明观测误差 的 ISS 性质（35），我们使用 ISS-Lyapunov 函数 ，并评估其导数
,
where and . Employing the inequality (19), and substituting , we get
在 和 的地方。利用不等式（19），并替换 ，我们得到
from which implies , and, hence, the ISS property (35) holds with the linear gain
其中 意味着 ，因此，ISS 性质（35）具有线性增益
.
We are now ready to design a control law as in Theorem 3. Noting that (41) and (42) is in strict feedback form, we apply one step of observer backstepping. For , we design the virtual control law . Denoting
我们现在准备按照定理 3 设计一个控制法则。注意到(41)和(42)处于严格反馈形式，我们应用一个观测器反步法的步骤。对于 ，我们设计虚拟控制法则 。记为
,
we rewrite as 我们将 重写为
.
The substitution of (47) in (52) yields , and, from (42) and (47),
(47)在(52)中的替换产生 ，并且，从(42)和(47)
,
where . Then, the control law
在 处。然后，控制法则
is implementable using the signals and , and results in
可使用信号 和 来实施，并导致
.
The remaining task is to select the design parameters and such that (39) and (40) are satisfied. For the ISSLyapunov function , the inequalities , (because ), , and , yield
剩下的任务是选择设计参数 和 ，使得(39)和(40)得到满足。对于 ISSLyapunov 函数 ，不等式 ， （因为 ）， ，以及 ，得到
We let , and select and to satisfy
我们让 ，并选择 和 来满足
,
so that 为了让
,
from which (40) follows for . For and , we now compute the gains and in (39). Using the fact that for each constant , for all , we obtain
从中得出 (40) 对于 。对于 和 ，我们现在计算 (39) 中的增益 和 。利用每个常数 ，我们得到所有 的 。
from which it follows that
由此可见
Then, (39) follows because , and the gains are
然后，(39)是因为 ，而且收益是
.
Using (44), (51) and (62), the inner and outer loop smallgain conditions, (36) and (37) are, respectively,
使用（44）、（51）和（62），内环和外环小增益条件，（36）和（37）分别是，
.
Selecting and sufficiently large ensures that (63) and (64) hold. Additional freedom for the selection of
选择足够大的 和 可以确保(63)和(64)成立。对于选择的额外自由
and is obtained from , which allocates the inner and outer loop gains.
和 是从 获得的，该参数分配了内环和外环增益。

5. Conclusion 5. 结论

Output-feedback control of nonlinear systems continues to be a challenging research area in which every new effort attempts to remove previous restrictions and improve robustness. The advance made in this paper is to allow the presence of nondecreasing nonlinearities to depend on linear combinations of unmeasured states. Global convergence of the nonlinear observer is achieved without the usual global Lipschitz restriction or highgain feedback. Instead, admissible nonlinearities and system structures are determined via verifiable feasibility conditions, while the design is carried out with computationally efficient LMI software. The ubiquitous question of observer and closed-loop system robustness is also answered for a significant class of modeling errors. It is hoped that these results will reinvigorate the search for new structure-specific nonlinear observer designs.
非线性系统的输出反馈控制仍然是一个具有挑战性的研究领域，每一次新的努力都试图消除先前的限制并提高鲁棒性。本文的进展在于允许非递减非线性取决于未测量状态的线性组合。非线性观测器的全局收敛性是在没有通常的全局 Lipschitz 限制或高增益反馈的情况下实现的。相反，通过可验证的可行性条件确定可接受的非线性和系统结构，同时设计是通过计算高效的 LMI 软件进行的。对于一类重要的建模误差，观测器和闭环系统鲁棒性的普遍问题也得到了解答。希望这些结果能够重新激发对新的结构特定非线性观测器设计的探索。

Appendix A 附录 A

A.1. Proof of Theorem 2
定理 2 的证明

Suppose the full-order observer LMI (10) is feasible for the system (20). Partitioning and as
假设全序观测器 LMI（10）对系统（20）是可行的。将 和 划分为
and substituting 替代
,
and  和
in (10), we obtain 在（10）中，我们获得
Defining , and , we note that (A.3) and (A.4) imply
定义 和 ，我们注意到(A.3)和(A.4)意味着
,
which is the reduced-order observer LMI (25). Suppose (69) and (70) hold for the system (20). To prove the existence of and satisfying (10), we rewrite system (20) in and coordinates, so that
这是降阶观测器 LMI（25）。假设系统（20）满足（69）和（70）。为了证明存在 和 满足（10），我们将系统（20）重写为 和 坐标系，这样
,
where and . We let
在 和 的地方。我们让
and obtain 并获得
In view of (A.5) and (A.6), the choice
鉴于（A.5）和（A.6），选择
results in the full-order observer LMI (10).
结果是全序观测器 LMI（10）。

A.2. Proof of finite escape time in system (30)
系统（30）中有限逃逸时间的证明

To prove that system (30) exhibits finite escape time for any choice of and , we use a variant of Lemma 3 from Mazenc, Praly, and Dayawansa (1994):
为了证明系统（30）对于任意选择的 和 都表现出有限逃逸时间，我们使用了 Mazenc、Praly 和 Dayawansa（1994）的引理 3 的一个变体：
Lemma 1. Consider the system
引理 1.考虑系统
where , and . If there exists a nonempty open set , and positive constants and such that, for all and ,
在 ，和 。如果存在一个非空开集 ，以及正常数 和 ，使得对于所有 和 ，
then, for and sufficiently large escapes to infinity in finite time.
然后，对于 和足够大的 ，在有限时间内逃逸到无穷远。
With the new variables and , system (30) is transformed into the
使用新变量 和 ，系统（30）被转换为
form (A.12) with 表格（A.12）与
Noting that the leading term of is , it is not difficult to show that the hypotheses of Lemma 1 hold with on any compact time interval in which is negative. Thus, with , and sufficiently large, the closed-loop system (30) exhibits finite escape time.
注意到 的主导项是 ，不难证明引理 1 的假设在任何紧凑时间间隔 上成立，其中 为负。因此，当 和 足够大时，闭环系统（30）表现出有限逃逸时间。

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Petar V. Kokotovic has been active for more than 30 years as control engineer, researcher and educator, first in his native Yugoslavia and then, from 1966 through 1990, at the University of Illinois, where he held the endowed Grainger Chair. In 1991 he joined the University of California, Santa Barbara where he directs the Center for Control Engineering and Computation. He has co-authored eight books and numerous articles contributing to sensitivity analysis, singular perturbation methods, and robust adaptive and nonlinear control. Professor Kokotovic is also active in industrial applications of control theory. As a consultant to Ford he was involved in the development of the first series of automotive computer controls and at General Electric he participated in large scale systems studies. Professor Kokotovic is a Fellow of IEEE, and a member of National Academy of Engineering, USA. He received the 1983 and 1993 Outstanding IEEE Transactions Paper Awards and presented the 1991 Bode Prize Lecture. He is the recipient of the 1990 IFAC Quazza Medal and the 1995 IEEE Control Systems Award.
Petar V. Kokotovic 在控制工程师、研究员和教育工作者方面已经活跃了 30 多年，最初在他的祖国南斯拉夫，然后从 1966 年到 1990 年在伊利诺伊大学担任捐赠的格林格椅子。 1991 年，他加入了加利福尼亚大学圣巴巴拉分校，负责控制工程和计算中心。他合著了八本书和大量文章，涉及灵敏度分析、奇异摄动方法以及强健自适应和非线性控制。Kokotovic 教授还积极参与控制理论在工业应用中的工作。作为福特的顾问，他参与了第一批汽车电脑控制系统的开发，并在通用电气公司参与了大规模系统研究。Kokotovic 教授是 IEEE 的会士，也是美国工程院的成员。他获得了 1983 年和 1993 年杰出 IEEE 交易论文奖，并发表了 1991 年博德奖演讲。他获得了 1990 年 IFAC Quazza 奖章和 1995 年 IEEE 控制系统奖。
Murat Arcak was born in Istanbul, Turkey in 1973. He received his B.S. degree in Electrical and Electronics Engineering from the Bogazici University, Istanbul in 1996, and his M.S. and Ph.D. degrees in Electrical and Computer Engineering from the University of California, Santa Barbara under the direction of Petar Kokotovic in 1997 and 2000, respectively. After a six-month post-doctoral period at the University of California, Santa Barbara, in January 2001 he joined the faculty at the Rensselaer Polytechnic Institute, Troy, NY, where he is currently an assistant professor in the Electrical, Computer and Systems Engineering Department. His research interests include all areas of nonlinear control theory and applications.
Murat Arcak 于 1973 年出生在土耳其伊斯坦布尔。他于 1996 年从伊斯坦布尔博阿齐奇大学获得电气与电子工程学士学位，并分别于 1997 年和 2000 年在加利福尼亚大学圣巴巴拉分别在 Petar Kokotovic 的指导下获得电气与计算机工程硕士和博士学位。在加利福尼亚大学圣巴巴拉进行了为期六个月的博士后研究后，他于 2001 年 1 月加入纽约特洛伊雷恩塞勒理工学院的教职，目前是该学院电气、计算机和系统工程系的助理教授。他的研究兴趣包括非线性控制理论和应用的所有领域。
  1. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Zhihua Qu under the direction of Editor Hassan Khalil. This research was supported in part by the National Science Foundation under Grant ECS-9812346 and the Air Force Office of Scientific Research under Grant F49620-95-1-0409.
    本文未在任何 IFAC 会议上发表。本文经副编辑 Zhihua Qu 推荐，在编辑 Hassan Khalil 的指导下以修订形式发表。本研究部分得到国家科学基金会 Grant ECS-9812346 和空军科学研究办公室 Grant F49620-95-1-0409 的支持。
    • Corresponding author. Tel.: + 1-518-276-6535; fax: + 1-518-2766261 .
      通讯作者。电话：+1-518-276-6535；传真：+1-518-2766261。
    E-mail addresses: arcak@ecse.rpi.edu (M. Arcak), petar@ece. ucsb.edu (P. Kokotović).