Static feedback stabilization of nonlinear systems with single sensor and single actuator

ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Research Article

Static feedback stabilization of nonlinear systems with single sensor and single actuator Jiqiang Wang n, Zhongzhi Hu, Zhifeng Ye College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing 210016, China

art ic l e i nf o

a b s t r a c t

Article history: Received 2 October 2012 Received in revised form 26 August 2013 Accepted 4 September 2013 This paper was recommended for publication by Prof. A.B. Rad.

This paper considers a single sensor and single actuator approach to the static feedback stabilization of nonlinear systems. This is essentially a remote control problem that is present in many engineering applications. The proposed method solves this problem that is less expensive to implement and more reliable in practice. Significant results are obtained on the design of controllers for stabilizing the nonlinear systems. Important issues on control implementation are also discussed. The proposed design method is validated through its application to nonlinear control of aircraft engines. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Static feedback stabilization Remote control Single sensor Single actuator

1. Introduction Any dynamic output feedback problem can be transformed into a static output feedback problem [1]. Static output feedback design is not only theoretically important, but also practically appealing due to its simplicity and reliability for implementation. However the problem of static output feedback control is challenging and still remains open, even for linear systems. This famous open problem in linear control theory can be stated as finding a static gain matrix K such that u ¼Ky stabilizes the linear system x_ ¼ Ax þ Bu, y ¼Cx, e.g. all the eigenvalues of the matrix (Aþ BKC) have strictly negative real parts. This deceptively simple problem is in fact complicated (even NP-hard) as minimization of spectral radius is known to be neither convex nor locally Lipchitz [2,3]. As a consequence, efforts are devoted to sufficient conditions for stability and the corresponding efficient algorithms for realization, e.g. q-SNM method for control with [4] and without constraints [5]; gradient sampling/non-smooth optimization technique [6]; multiobjective optimization method [7] etc. A literature survey shows that static output feedback control for nonlinear systems is not as widely investigated as its linear counterpart (see the recent paper [8]). For nonlinear systems, the nonconvexity issues persist, and no general result is available. However there are still advances in both leading to sufficient conditions and developing efficient algorithms for specific problems. For example,

n

Corresponding author. Tel.: þ 86 152 4021 1717; fax: þ 86 258 489 3666. E-mail addresses: [email protected], [email protected] (J. Wang).

the static output feedback stabilization problem is examined by an iterative sums of squares approach in [8], based on a semi-definite programming (SDP) method to solve state-dependent LMIs [9–11]. In [12] and [13], the problem is converted into the solvability of Hamilton–Jacobi equation, enabling approximation methods to obtain solutions (hence inherently suboptimal, see [14] and references therein for techniques to solve the Hamilton–Jacobi equation); a computational scheme of solving the static output feedback control problem for a class of polynomial nonlinear systems is proposed in [15], also exploring the sum of squares decomposition based methodology. In this paper, however, a Lyapunov function-based method is introduced for the design of static output feedback stabilization. The contributions can be summarized as follows: (1) it is different from the existing optimization-based approaches, hence not computationally demanding; (2) control action is restricted to one sensor and one actuator, henceforth to be the simplest possible implementation scheme, while all the results so far need more than one sensor to be implementable; (3) numerical studies are provided for control of aircraft engines, where the single sensor and single actuator approach is pretty novel. To understand comprehensively the significance of the second contribution, it is pointing out that this paper is partially motivated by the author's effort to controlling multiple performance variables using only restricted controls. This is a “remote” control problem where only one sensor signal is available to one local actuator, but performance is also required at remote locations. This point will be further delineated in the next section. This paper is structured as follows: Section 2 formulates the problem to be studied before presenting the main results in Section 3. While the results are applicable for disturbance-free case,

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.09.010

Please cite this article as: Wang J, et al. Static feedback stabilization of nonlinear systems with single sensor and single actuator. ISA Transactions (2013), http://dx.doi.org/10.1016/j.isatra.2013.09.010i

J. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

control design with bounded disturbances is addressed in Section 4. Section 5 provides a numerical study for validation of the proposed design, and some important issues associated with the design methodology are also discussed. Finally Section 6 concludes the paper.

ð5Þ

2

Furthermore, g1(z1,z2) being Lipchitz implies there can be found another positive number λ4 such that: J g 1 ðz1 ; z2 Þ J r λ4 J z1 J

2. Problem formulation

ð6Þ

The main results of the paper can now be stated:

Consider a general nonlinear system x_ ¼ f ðxÞ that is rewritten into the following form: z_ 1 ¼ g 1 ðz1 ; z2 Þ; z_ 2 ¼ g 2 ðz1 ; z2 Þ

ð1Þ

where: the state variable z1 is a scalar; the state variable z2 ¼ ð x1 ⋯ xN  1 ÞT is a vector, with the subsystem z_ 2 ¼ g 2 ðz1 ; z2 Þ being Lipchitz in a neighbourhood of z1 ¼0 while z_ 2 ¼ g 2 ð0; z2 Þ being uniformly exponentially stable about z2 ¼0 8 z. The output equation is y¼h(z1, z2) with h being nonlinear function of the states. The objective is to stabilize the system (1) using only static and single input: u ¼ ky;



∂V 0 ðz2 Þ

∂z r λ3 J z2 J

Theorem 1. Consider the following single-input nonlinear control system: z_ 1 ¼ g 1 ðz1 ; z2 Þ þu1 ; z_ 2 ¼ g 2 ðz1 ; z2 Þ If the following condition holds: z1 g 1 ðz1 ; z2 Þ þ z1 kðz1 ; z2 Þ þ

That is to say, the system should be stabilized using a single sensor and a single actuator. As stated above, this is an important problem that is often overlooked in practical designs since, on one hand, actuators are expensive and/or heavy and are hence avoided in a system design [16]; on the other hand, there are many situations where in-service information from z2 is simply either not obtainable or prohibitively expensive to do so. As a consequence, control design can only proceed with z1 (often remote to z2) and its feedback action, but with the performance objective of controlling both z1 and z2, refer to Fig. 1. Such examples are abounding, particularly in the interconnected large scale structures. For example, in the helicopter or submarine rotor blade control, only the shaft acceleration is available for feedback, while both the shaft vibration and the blade vibration should be attenuated. Obviously here the difficulty is that it is not practically viable to permanently locate sensors into the blades or prohibitively expensively to do so. Although there exist approaches such as integrating smart materials into the blades, these solutions are costly and unproven in real operational environments. The research presented here should be considered as an ideal alternative solution to the remote control problem [17].

ð8Þ

cVðz1 ; z2 Þ 1 ; Vðz1 ; z2 Þ ¼ z21 þ V 0 ðz2 Þ z1 2 u1 ¼ 0 if z1 ¼ 0

u1 ¼ kðz1 ; z2 Þ  z1 a 0

if ð9Þ

exponentially stabilizes the closed loop system. Proof Consider the Lyapunov function candidate Vðz1 ; z2 Þ ¼ ð1=2Þz21 þ V 0 ðz2 Þ. Differentiating it along the trajectory of (6) leads to: _ 1 ; z2 Þ ¼ z1 z_ 1 þ ∂V 0 ðz2 Þg ðz1 ; z2 Þ Vðz 2 ∂z2   cVðz1 ; z2 Þ ∂V 0 ðz2 Þ ¼ z1 g 1 ðz1 ; z2 Þ þ kðz1 ; z2 Þ  g 2 ðz1 ; z2 Þ þ z1 ∂z2 ¼ z1 g 1 ðz1 ; z2 Þ þ z1 kðz1 ; z2 Þ  cVðz1 ; z2 Þ þ

∂V 0 ðz2 Þ g 2 ðz1 ; z2 Þ ∂z2

r  cVðz1 ; z2 Þ Hence the nonlinear control system (7) is guaranteed to be exponentially stabilized. Theorem 2. For the nonlinear control system (7), the single input and static state feedback control cVðz1 ; z2 Þ z1 if z1 ¼ 0

u1 ¼ k1 z1  3. Main results

∂V 0 ðz2 Þ g 2 ðz1 ; z2 Þ r 0 ∂z2

where k(z1,z2) is a polynomial in z1 and z2, then the single-input and static state feedback control

ð2Þ

k is a constant

ð7Þ

u1 ¼ 0

if

z1 a 0 ð10Þ

To state the main result, first notice that the assumption that the subsystem z_ 2 ¼ g 2 ðz1 ; z2 Þ is Lipchitz in a neighbourhood of z1 ¼ 0 implies, there can be found a positive constant λ1 such that:

exponentially stabilizes the closed loop system, provided the static gain k1 satisfies the following condition:

J g 2 ðz1 ; z2 Þ  g 2 ð0; z2 Þ J r λ1 J z1 J

k1 r  λ4 

ð3Þ

λ21 λ23 4λ2

ð11Þ

Also 8 z, the assumption that z_ 2 ¼ g 2 ð0; z2 Þ is uniformly exponentially stable about z2 ¼ 0 implies there exist a Lyapunov function V0(z2) and two positive numbers λ2, λ3 such that:

Proof. In condition (8), substitute k(z1,z2) ¼k1z1, the following is obtained:

∂V 0 ðz2 Þ V_ 0 ðz2 Þ ¼ g 2 ð0; z2 Þ r  λ2 J z2 J 2 ∂z2

z1 g 1 ðz1 ; z2 Þ þ k1 J z1 J 2 þ

w u

P

ð4Þ

z2

or:

z1

z1 g 1 ðz1 ; z2 Þ þ k1 J z1 J 2 þ þ

k Fig. 1. Static feedback control using single sensor and single actuator. This is actually a remote control problem [17].

∂V 0 ðz2 Þ g 2 ðz1 ; z2 Þ r 0 ∂z2

ð12Þ

∂V 0 ðz2 Þ ½g 2 ðz1 ; z2 Þ  g 2 ð0; z2 Þ ∂z2

∂V 0 ðz2 Þ g 2 ð0; z2 Þ r0 ∂z2

ð13Þ

Hence, from the conditions (3)–(6): z1 g 1 ðz1 ; z2 Þ r λ4 J z1 J 2 ;

∂V 0 ðz2 Þ ½g 2 ðz1 ; z2 Þ  g 2 ð0; z2 Þ r λ1 λ3 J z1 J J z2 J ∂z2

Please cite this article as: Wang J, et al. Static feedback stabilization of nonlinear systems with single sensor and single actuator. ISA Transactions (2013), http://dx.doi.org/10.1016/j.isatra.2013.09.010i

J. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Therefore the condition (13) will be satisfied if the following inequality stands: λ4 J z1 J 2 þ k1 J z1 J 2 þ λ1 λ3 J z1 J J z2 J  λ2 J z2 J 2 r 0

ð14Þ

For any z1 and z2, condition (14) will hold if λ21 λ23 þ 4λ2 ðk1 þ λ4 Þ r 0 and this gives the condition (11). To recap, if the static gain k1 is such chosen that satisfies condition (11) k1 r  λ4  λ21 λ23 =4λ2 , then the Lyapunov function candidate Vðz1 ; z2 Þ ¼ ð1=2Þz21 þ V 0 ðz2 Þ along the trajectory of the _ r  cVðtÞ. Thus expononlinear control system (7) satisfies VðtÞ nential stabilization is achieved via the single input, static state feedback control (10). Remark 1. Comparing with the controller (9), the controller (10) uses only a static gain k(z1,z2)¼ k1z1, thus simplifies control implementation. The static gain k1 is only constrained by the condition (11). However, the controller (10) is still “complex” as its implementation requires sensing of both z1 and z2. As is pointed out above, there are many situations where in-service information from z2 is either not obtainable or cost-effective. It is therefore desirable to design a controller based only on z1, whiling controlling both z1 and z2. This important problem is solved in the following result. Theorem 3. For the nonlinear control system (7), if the static gain k1 satisfies k1 r  λ4 

λ21 λ23 4λ2

ð15Þ

cjz1 j if z1 if z1 ¼ 0

u1 ¼ k1 z 1  u1 ¼ 0

ð16Þ

asymptotically stabilizes the closed loop system. Proof. Consider again the Lyapunov function candidate Vðz1 ; z2 Þ ¼ ð1=2Þz21 þ V 0 ðz2 Þ and differentiate it along the trajectory of (6): _ 1 ; z2 Þ ¼ z1 z_ 1 þ ∂V 0 ðz2 Þg ðz1 ; z2 Þ Vðz 2 ∂z2   cjz1 j ∂V 0 ðz2 Þ g 2 ðz1 ; z2 Þ þ ¼ z1 g 1 ðz1 ; z2 Þ þk1 z1  z1 ∂z2

þ

Any realistic system consists of disturbances and/or noises in one form or another, and consequently the corresponding control design must possess certain robustness property. This section considers the problem of static feedback control of nonlinear systems with bounded disturbance. Consider the following singleinput control system: z_ 1 ¼ g 1 ðz1 ; z2 Þ þ u1 þ d1 ðtÞ; z_ 2 ¼ g 2 ðz1 ; z2 Þ þ d2 ðtÞ

ð17Þ

where: d1(t), d2(t) are disturbances entering the two subsystems, satisfying ||d1(t)|| rl1 and ||d2(t)|| rl2, respectively. The problem is to stabilize the system (16) using a single sensor and a single actuator. This problem is addressed in the following important result: Theorem 4. For the nonlinear control system (16), if the static gain k1 satisfies k1 r  λ4 

λ21 λ23 4λ2

ð18Þ

then the single input and static state feedback control cVðz1 ; z2 Þ z1 if z1 ¼ 0

u1 ¼ k1 z1  u1 ¼ 0

if

z1 a0 ð19Þ

exponentially stabilizes the closed loop system to the region Q defined by: z1 Z 0;

z2 Z 0;

ðλ4 þ k1 Þ J z1 J 2

þλ1 λ3 J z1 J J z2 J  λ2 J z2 J 2 þ l2 λ3 J z2 J þ l1 J z1 J ¼ 0g

z1 a 0

¼ z1 g 1 ðz1 ; z2 Þ þ k1 ‖z1 ‖2 þ

4. Control design with bounded disturbance

Q ¼ f jðz1 ; z2 Þj

then the single input, single state feedback control

3

 ∂V 0 ðz2 Þ g 2 ðz1 ; z2 Þ  g 2 ð0; z2 Þ ∂z2

∂V 0 ðz2 Þ g 2 ð0; z2 Þ  cjz1 j ∂z2

r λ4 J z1 J 2 þ k1 J z1 J 2 þλ1 λ3 J z1 J J z2 J  λ2 J z2 J 2  cjz1 j Therefore from the proof of Theorem 2, the condition k1 r λ4 λ21 λ23 =4λ2 implies that the inequality λ4 J z1 J 2 þ k1 J z1 J 2 þ λ1 λ3 _ 1 ; z2 Þ r cjz1 j o 0 J z1 J J z2 J λ2 J z2 J 2 r 0 always holds. Hence Vðz 8 z1 a0. This proves the result. Remark 2. It is remarkable that the design parameter k1 in controller (10) and controller (15) satisfies the same inequality. However comparing with controller (10), controller (15) requires only information from z1, hence is the single sensor and single actuator approach to controlling the nonlinear system. For static output feedback control problem, this implies that the output equation becomes y ¼ hðz1 ; z2 Þ ¼ z1 þ c=k1 z1 . This is not really a handicap as most of the iterative sums of squares approaches for nonlinear static output feedback control only admit linear output equation y¼Cx (C is constant) to relieve the computational demand, see the numerical examples in [8].

Furthermore, for the same static gain k1 satisfying (11), the single input, single state feedback control u1 ¼ k1 z1  u1 ¼ 0

if

cjz1 j z1

if

z1 a0

z1 ¼ 0

ð20Þ

asymptotically stabilizes the closed loop system to Q. Proof. Consider the Lyapunov function candidate Vðz1 ; z2 Þ ¼ ð1=2Þ z21 þ V 0 ðz2 Þ and differentiate it along the trajectory of (16): _ 1 ; z2 Þ ¼ z1 z_ 1 þ ∂V 0 ðz2 Þ½g ðz1 ; z2 Þ þd2 ðtÞ Vðz 2 ∂z2   cVðz1 ; z2 Þ þ d1 ðtÞ ¼ z1 g 1 ðz1 ; z2 Þ þ k1 z1  z1 þ

∂V 0 ðz2 Þ ½g 2 ðz1 ; z2 Þ þ d2 ðtÞ ∂z2

¼ z1 g 1 ðz1 ; z2 Þ þ k1 J z1 J 2 þ þ z1 d1 ðtÞ þ

∂V 0 ðz2 Þ g 2 ðz1 ; z2 Þ ∂z2

∂V 0 ðz2 Þ d2 ðtÞ  cVðz1 ; z2 Þ ∂z2

r ðλ4 þ k1 Þ J z1 J 2 þ λ1 λ3 J z1 J J z2 J  λ2 J z2 J 2 þ l2 λ3 J z2 J þ l1 J z1 J  cVðz1 ; z2 Þ Consider Fðz1 ; z2 Þ  ðλ4 þk1 Þ J z1 J 2 þ λ1 λ3 J z1 J J z2 J  λ2 J z2 J 2 þ l2 λ3 J z2 J þ l1 J z1 J . After a change of coordinates, F(z1,z2) can be rewritten into: Fðz1 ; z2 Þ ¼ ðλ4 þ k1 Þð J z1 J  J z10 J Þ2 þ λ1 λ3 ð J z1 J  J z10 J Þð J z2 J  J z20 J Þ  λ2 ð J z2 J  J z20 J Þ2  ðλ4 þ k1 Þ J z10 J 2 þ λ2 J z20 J 2  λ1 λ3 J z10 J J z20 J

ð21Þ

Please cite this article as: Wang J, et al. Static feedback stabilization of nonlinear systems with single sensor and single actuator. ISA Transactions (2013), http://dx.doi.org/10.1016/j.isatra.2013.09.010i

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4

where ||z10|| and ||z20|| are determined by the method of undetermined coefficients: J z10 J ¼ 

J z20 J ¼

2l1 λ2 þ l2 λ1 λ23 4λ2 λ4 þ4k1 λ2 þ λ21 λ23

2l2 λ3 ðk1 þ λ4 Þ  l1 λ1 λ3 4λ2 λ4 þ 4k1 λ2 þλ21 λ23

5. Application and discussion: Nonlinear control of aircraft engines

ð22Þ

This subsection presents a realistic study on nonlinear control of aircraft engines. First consider the following nonlinear model around certain operating point without disturbance entering in:

ð23Þ

Δn_ 1 ¼  4:1476Δn1 þ 1:4108Δn2 þ 12Δn21  Δn22 þ Δwf Δn_ 2 ¼ 0:2975Δn1  3:1244Δn2  1:7Δn21 þ Δn22

Substitute into (17) and manipulate, it is seen that F(z1,z2) ¼0 is defined by the equation: ðλ4 þ k1 Þð J z1 J  J z10 J Þ2 þ λ1 λ3 ð J z1 J  J z10 J Þð J z2 J  J z20 J Þ  λ2 ð J z2 J  J z20 J Þ2 ¼  C

ð24Þ

2 1=4λ2 ½l2 λ23 ðð2l1 λ2 þ l2 λ1 λ23 Þ2 =ð4λ2 λ4 þ 4k1 λ2 þ λ21 λ23 ÞÞ. 

where: C ¼ Now let Q ¼ ð z1 ; z2 ; Fðz1 ; z2 Þ Þj z1 Z0; z2 Z 0; Fðz1 ; z2 Þ ¼ 0g, it is seen that for a specific design k1, Q encloses a domain on z1  z2 plane (Fig. 2) whose area depends on C, hencefore on the disturbance bounds l1 and l2, e.g. l1 ¼l2 ¼0, Q reduces to a single point at the origin (0, 0, 0). If now the static control gain is such chosen satisfying: λ21 λ23 þ 4λ2 ðk1 þ λ4 Þ r 0

ð25Þ

For any z1 and z2, there will have F(z1,z2) r0 outside Q on z1  z2 plane with equality obtained on the boundary. That is for (z1,z2) A ℜ2 Q: _ 1 ; z2 Þ r Fðz1 ; z2 Þ  cVðz1 ; z2 Þ r  cVðz1 ; z2 Þ Vðz

ð26Þ

The system will be exponentially stabilized to Q via the single input, static state feedback control (10). A similar argument can be used to show that the single input, single state feedback control (15) can asymptotically stabilize the system (16) to Q. Remark 3. Comparing with the disturbance free case, the static gain k1 can only provide stability to a region Q other than the origin. However it is noted that the enclosed area depends on the disturbance bounds l1 and l2, e.g. l1 ¼l2 ¼0, Q reduces to the origin (0, 0, 0); on the other hand, for given l1 and l2, a sufficiently small k1 will reduce Q, which can be demonstrated from the generalized Gronwall–Bellman lemma but nevertheless provide a guidance for design.

ð27Þ

where: Δn1 represents the rotational speed change in low pressure turbines while Δn2 represents the rotational speed change in high pressure turbines; Δwf is the fuel flow ratio as control input. The model is adopted from the Intelligent Engine Control project [18]. The nonlinear part has been augmented for the investigation of advanced control concepts, comparing with linear controls exclusively utilized in the state-of-art design. Now let z1 ¼ Δn1, z2 ¼Δn2, it is seen that the subsystem z_ 2 ¼ g 2 ðz1 ; z2 Þ is smooth in a neighbourhood of z1 ¼ 0 and z_ 2 ¼ g 2 ð0; z2 Þ is uniformly exponentially stable about z2 ¼0 8 z. The assumption made in this paper is satisfied. Take again Vðz1 ; z2 Þ ¼ ð1=2Þz21 þ V 0 ðz2 Þ ¼ ð1=2Þðz21 þ z22 Þ. From Theorem 1, it is known that the controller (9) can exponentially stabilize the system (22) if the polynomial k(z1,z2) is such chosen that the following condition is satisfied: z1 g 1 ðz1 ; z2 Þ þ z1 kðz1 ; z2 Þ þ z2 g 2 ðz1 ; z2 Þ r 0

ð28Þ

Now choose the static gain k(z1,z2)¼ k1z1, and substitute (22) into (23) leads to: ðk1  4:1476  1:7z2 þ 12z1 Þz21 þ 1:7083z1 z2 þðz2  z1  3:1244Þ r 0 ð29Þ By the technique of completing the squares, together with the consideration of |Δn1| r1, |Δn2| r 1, it is obtained that inequality (24) will hold if the static gain k1 satisfies: k1 o  10:2073

ð30Þ

Take k1 ¼ 11 and the single-input and static state feedback control is given by: u1 ¼  11z1 

cVðz1 ; z2 Þ z1

ð31Þ

Theorems 1, 2 and the above analysis guarantee that controller (26) will asymptotically stabilize the engine (22). A simulation with initial condition ½ z1 ð0Þ z2 ð0Þ  ¼ ½  0:5 1:0 , c¼ 2 confirms the assertion, see Figs. 3 and 4, also shown is the performance of the PI control for benchmarking.

F(z1,z2)=0 1

1.5

1

n1: proposed control n1: without control n1: PI control n2: proposed control n2: without control n2: PI control

z2

Output Signal

0.5

0

0.5

0

-0.5 0

0.2

0.4

0.6

z1 Fig. 2. An example of the domain Q.

0.8

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s) Fig. 3. Regulation of the aircraft engine via single input and static state feedback.

Please cite this article as: Wang J, et al. Static feedback stabilization of nonlinear systems with single sensor and single actuator. ISA Transactions (2013), http://dx.doi.org/10.1016/j.isatra.2013.09.010i

J. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 8

16

proposed control signal PI control signal

14

5

Control Signal

10

Control Signal

6

8 6

4 3

4

2

2

1

0

0

0

0.1

0.2

0.3

0.4

proposed control signal PI control signal

7

12

-2

5

-1

0.5

0

0.2

0.4

0.6

0.8

Time (s)

1

1.2

1.4

1.6

1.8

2

Time (s)

Fig. 4. Response of the control signal u1 in (26).

Fig. 6. Response of the control signal u1 in (27).

1 n1: proposed control n1: without control n1: PI control n2: proposed control n2: without control n2: PI control

Output Signal

0.5

1 n1: proposed control n1: without control n1: PI control n2: proposed control n2: without control n2: PI control

Output Signal

0.5

0

0

-0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 -0.5

Time (s) Fig. 5. Regulation of the aircraft engine via single input and single actuatordisturbance free.

0

0.2

0.4

0.6

0.8

1

1.2

if

1.8

2

8 proposed control signal PI control signal

7

cjz1 j z1

1.6

Fig. 7. Regulation of the aircraft engine via single input and single actuator with disturbances.

The single sensor and single actuator solution can now be obtained from Theorem 3, which asserts that for the same control gain k1, the following controller can provide asymptotical stability: u1 ¼ 11z1 

1.4

Time (s)

z1 a 0

6

if

z1 ¼ 0

ð32Þ

It is again observed that the asymptotic stabilizer leads to a relatively sluggish response than the exponential one, but the control effort is reduced significantly. See Figs. 5 and 6. Now consider the case where disturbances affect both Δn1 and Δn2 with d1 ðtÞ ¼ 2:5 sin ð60tÞ þ 0:05Uð0; 1Þ and d2 ðtÞ ¼ 1:5 sin ð80tÞ þ0:1Uð0; 1Þ. To be more general, both disturbances consist of a random noise superposed by a low frequency signal, representing typical pulsation dynamics of the engine fuel flow regulator. According to Theorem 4, for the same static gain k1, the single input, single state feedback control (27) will asymptotically stabilize the closed loop system to a region Q as shown in Figs. 7 and 8. It is seen that the proposed design provides much better performance than that of PI control. Even remarkable is that almost asymptotical stability to origin for Δn1 is achieved in spite of disturbances while this is not the case for PI control that responds violently to the disturbances.

Control Signal

5

u1 ¼ 0

4 3 2 1 0 -1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s) Fig. 8. Response of the control signal u1 in (27) and PI control.

Remark 4. Simulations show that the regulation performance of Δn1 is much better than that of Δn2. This is so since the low pressure turbine has only aerodynamic coupling with the high

Please cite this article as: Wang J, et al. Static feedback stabilization of nonlinear systems with single sensor and single actuator. ISA Transactions (2013), http://dx.doi.org/10.1016/j.isatra.2013.09.010i

J. Wang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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pressure turbine. However it is exactly due to this aerodynamic coupling that prevents the direct control of Δn1 and Δn2 simultaneously, necessitating the single actuator approach to the control of aircraft engines. Remark 5. The state of the art in aircraft engine control utilizes linear design methodologies exclusively. Advanced nonlinear control can accommodate large plant variation over the flight envelope, hence representing one of the important research directions in the field. The results presented here contribute to this important area of research as well. 6. Conclusion The problem of single sensor and single actuator stabilization of nonlinear systems has been discussed. Some sufficient conditions have been obtained and these theoretical results have been validated through the nonlinear regulation of aircraft engines. Different from the existing optimization-based approaches, the proposed design method is relatively easy and non-computationally demanding. Even important is the single sensor and single actuator approach for nonlinear control that can be very desirable for practical engineering systems. In fact, the method proposed here solves the challenging remote control problem that has been encountered in diverse areas of applications. Acknowledgements This work is supported by the NUAA Fundamental Research Funds (NS2013020). References [1] Syrmos VL, Abdallah CT, Dorato P, Grigoriadis K. Static output feedback: a survey. Automatica 1997;33:125–37.

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Please cite this article as: Wang J, et al. Static feedback stabilization of nonlinear systems with single sensor and single actuator. ISA Transactions (2013), http://dx.doi.org/10.1016/j.isatra.2013.09.010i