Stabilizability and Disturbance Rejection with State-Derivative Feedback

Abstract

In some practical problems, for instance in the control of mechanical systems using accelerometers as sensors, it is easier to obtain the state-derivative signals than the state signals. This paper shows that (i) linear time-invariant plants given by the state-space model matrices {𝐴,𝐵,𝐶,𝐷} with output equal to the state-derivative vector are not observable and can not be stabilizable by using an output feedback if det(𝐴)=0 and (ii) the rejection of a constant disturbance added to the input of the aforementioned plants, considering det(𝐴)≠0, and a static output feedback controller is not possible. The proposed results can be useful in the analysis and design of control systems with state-derivative feedback.

1. Introduction

There exist many results in the literature on the feedback control of systems described through state variables [1]. State feedback or output feedback is usually used, but in some cases the state-derivative feedback can be very useful to achieve a desired performance [2].

In the last years, the state-derivative feedback of linear systems has been studied by some researchers. In [3], the authors proposed a formula similar to the Ackermann formula, for the pole-placement design with a state-derivative feedback gain. In [4, 5], a linear quadratic regulator (LQR) controller design scheme for standard state-space systems was presented. The procedure described in [6] allows the design of state-derivative feedback control systems using state feedback design methods. Other results about the pole placement of multivariable system, with state-derivative feedback, can be found in [7–13].

There exist some practical problems where the state-derivative signals are easier to obtain than the state signals, for instance, in the control of mechanical systems for the suppression of vibration, where the main sensors are accelerometers. In this case, from the signals of the accelerometers it is possible to reconstruct the velocities with a good precision but not the displacements [3]. Defining the velocities and displacements as the state variables, one has available feedback for the state-derivative signals. Because of their low cost, accelerometers have been used in the industry, for instance, in the following applications: suppression of vibration in mechanical systems [3, 14, 15], control of car wheel suspension systems [16], aeronautical engineering [5, 15], and civil engineering [4]. Recently, new LMI-based state-derivative feedback design methods have been presented [17, 18]. These design methods allow new specifications and also the consideration of a broader class of plants than the related results available in the literature [3, 4, 7, 16–19].

This paper investigates the observability, stabilizability, and disturbance rejection of linear time-invariant plants given by the state-space model matrices {𝐴,𝐵,𝐶,𝐷}, with output equal to the state-derivative vector. Section 2 shows that if det(𝐴)=0, then the aforementioned plants are not observable, are unstable (because they have at least one pole equal to zero), and cannot be stabilizable by using output feedback. The assumption det(𝐴)≠0 has been considered in the linear state-derivative designs [3, 6, 7, 17–20]. The proposed results in Section 2 elucidate and give more details about the condition det(𝐴)=0. The necessity of the condition det(𝐴)≠0, considering the state-derivative feedback, is a new result offered in this paper. Section 3 shows that if det(𝐴)≠0, then the plants {𝐴,𝐵,𝐶,𝐷} cited above present at least one transmission zero equal to zero. Furthermore, due to this fact, as described in Section 3, the disturbance rejection of a constant disturbance added to the plant input is not possible, by using a static state-derivative feedback. This conclusion does not invalidate the application of the state-derivative feedback controllers, because the main use of these controllers is in the suppression of vibrations, where the disturbances usually have a mean value equal to zero. This paper is an expanded and updated version of [21]. Other results about the stabilizability and stability robustness of state-derivative feedback, including the fragility, can be found in [22].

2. Observability and Stabilizability of Systems with State-Derivative Feedback

In the control of linear time-invariant systems, represented by the state-space model matrices {𝐴,𝐵,𝐶,𝐷}, the design of state-derivative feedback usually supposes that det(𝐴)≠0 [3, 6, 7, 17–19]. In this paper this condition is investigated with details and related with the observability and stability of this system. The main idea in this study was the representation of the state-derivative vector 𝑥̇(𝑡) as the output of this system, 𝑦(𝑡).

Lemma 2.1. Consider the linear time-invariant system, with input 𝑢(𝑡) and output 𝑦(𝑡) described by 𝑥̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)=𝑥̇(𝑡)(𝑡)(𝑡)(𝑡)(𝑡),=𝐴𝑥+𝐵𝑢=𝐴𝑥+𝐵𝑢=𝐶𝑥+𝐷𝑢 (2.1) where 𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑚, and 𝑥(𝑡)∈ℝ^𝑛×1.
Then, this system is observable if and only if det(𝐴)≠0.

Proof. Note that the matrices of the system (2.1) are the following: {𝐴,𝐵,𝐶,𝐷}={𝐴,𝐵,𝐴,𝐵}. The observability matrix of the system (2.1) is given by [23] 𝐶𝒪=𝐶𝐴𝐶𝐴²⋮𝐶𝐴^𝑛−1=𝐴𝐴²𝐴³⋮𝐴^𝑛=𝐼_𝑛𝐴𝐴²⋮𝐴^𝑛−1𝐴. (2.2) Then, from (2.2), it follows from the Sylvester inequality [23] that (𝒪)𝐼rank=rank^𝑛𝐴𝐴²⋮𝐴^𝑛−1𝐴=rank(𝐴)=𝑛 (2.3) if and only if det(𝐴)≠0. Thus, the system (2.1) is observable if and only if det(𝐴)≠0.

The next lemma presents results about the poles of the system (2.1), when det(𝐴)=0.

Lemma 2.2. Suppose that det(𝐴)=0 and rank(𝐴)=𝑞<𝑛. Then, the system (2.1) can be decomposed, by an equivalent transformation, into observable and unobservable parts, and the unobservable part has 𝑛−𝑞 poles at 𝑠=0.

Proof. Since det(𝐴)=0 and rank(𝐴)=𝑞<𝑛, the matrix 𝐴 can be represented as follows: 𝐴=𝑅𝐴₀, (2.4) where 𝑅∈ℝ^𝑛×𝑞, 𝐴₀∈ℝ^𝑞×𝑛, and rank(𝐴₀)=𝑞. Define 𝐴₀₁∈ℝ^{𝑛−𝑞×𝑛},rank(𝐴₀₁)=𝑛−𝑞, and 𝐴₀𝐴^𝑇₀₁=0. Then, performing the linear transformation 𝐴𝑧=𝑇𝑥,𝑇=⁰¹𝐴₀𝐴^𝑇₀⁻¹𝐴₀, (2.5) the system (2.1) can be transformed into 𝑧̇=𝑇𝐴𝑇⁻¹𝑧+𝑇𝐵𝑢,𝑦=𝐴𝑇⁻¹𝑧+𝐵𝑢, (2.6) where 𝑇⁻¹=𝐴^𝑇₀₁𝐴₀₁𝐴^𝑇₀₁⁻¹𝐴^𝑇₀. (2.7) From (2.4)–(2.7) note that 𝐴𝑇⁻¹=𝑅𝐴₀𝐴^𝑇₀₁𝐴₀₁𝐴^𝑇₀₁⁻¹𝐴^𝑇₀=0𝑅𝐴₀𝐴^𝑇₀. (2.8) Thus, from (2.6)–(2.8) and defining 𝑧^𝑇=[𝑧^𝑇_𝑎𝑧^𝑇_𝑏],𝑧_𝑎∈ℝ^𝑛−𝑞, and 𝑧_𝑏∈ℝ^𝑞, one has 𝑦=𝐴𝑇⁻¹𝑧+𝐵𝑢=𝑅𝐴₀𝐴^𝑇₀𝑧_𝑏+𝐵𝑢. (2.9) Now, from (2.4)–(2.7) note that ̇𝑧̇𝑧=^𝑎𝑧̇_𝑏=𝐴₀₁𝐴₀𝐴^𝑇₀⁻¹𝐴₀𝑅𝐴₀𝐴₀₁𝐴₀𝐴^𝑇₀⁻¹𝐴₀^𝑇𝑧_𝑎𝑧_𝑏+𝐴₀₁𝐴₀𝐴^𝑇₀⁻¹𝐴₀=𝐵𝑢0𝐴₀₁𝑅𝐴₀𝐴^𝑇₀0𝐴₀𝐴^𝑇₀⁻¹𝐴₀𝑅𝐴₀𝐴^𝑇₀𝑧_𝑎𝑧_𝑏+𝐴₀₁𝐵𝐴₀𝐴^𝑇₀⁻¹𝐴₀𝐵𝑢. (2.10) Hence, observe that 𝑧_𝑎 cannot be detected from the output 𝑦, because 𝑧_𝑎 is not connected to 𝑦 in (2.9) and also to 𝑧̇_𝑏 in (2.10). Therefore the subsystem with state vector 𝑧_𝑎 is not observable. Now, from (2.10) it follows that 𝑧̇_𝑎=𝐴₀₁𝑅𝐴₀𝐴^𝑇₀𝑧_𝑏+𝐴₀₁𝐵𝑢, (2.11) and this system has 𝑛−𝑞 poles at 𝑠=0. This completes the proof of Lemma 2.2.

The next lemma addresses the stabilization of system (2.1).

Lemma 2.3. If det(𝐴)=0, the system (2.1) cannot be stabilizable by using an output feedback.

Proof. From Lemma 2.2, if det(𝐴)=0 and rank(𝐴)=𝑞<𝑛, then the system (2.1) is unobservable and the unobservable part has 𝑛−𝑞 poles in 𝑠=0. Thus, the unobservable part is not connected to the output and it is not possible to change its poles by using an output feedback. Therefore, the stabilization with an arbitrary output feedback is not possible. The lemma is proved.

3. Disturbances Rejection with State-Derivative Feedback

Results on the rejection of constant disturbances, added to the input of the plant (2.1), considering the state-derivative feedback are presented. Initially, the next lemma shows that plant (2.1) has at least one transmission zero at 𝑠=0 if det(𝐴)≠0.

Lemma 3.1. If det(𝐴)≠0, then the system (2.1) has at least one transmission zero at 𝑠=0.

Proof. Consider the system (2.1), given by 𝑥̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)=𝑥̇(𝑡).=𝐴𝑥+𝐵𝑢 (3.1) Applying the Laplace transform and considering the initial conditions equal to zero, then (𝑠)(𝑠)(𝑠),(𝑠)(𝑠)(𝑠),()𝑋(𝑠)(𝑠),𝑋(𝑠)=()𝑠𝑋=𝐴𝑋+𝐵𝑈𝑠𝑋−𝐴𝑋=𝐵𝑈𝑠𝐼−𝐴=𝐵𝑈𝑠𝐼−𝐴⁻¹(𝑠),𝑌(𝑠)(𝑠).𝐵𝑈=𝑠𝑋 (3.2) From (3.2) it follows that 𝑌(𝑠)=𝑠(𝑠𝐼−𝐴)⁻¹𝐵𝑈(𝑠). (3.3) Thus, supposing that det(𝐴)≠0, then 𝐴⁻¹ can be calculated and 𝑠(𝑠𝐼−𝐴)⁻¹𝐵_𝑠=0=−𝑠𝐴⁻¹𝐵_𝑠=0=0. (3.4) This completes the proof of the existence of transmission zeros at 𝑠=0.

The existence of transmission zeros at 𝑠=0, as demonstrated in Lemma 3.1, is a problem in the rejection of constant disturbances added to the plant input with a state-derivative feedback controller. Lemma 3.2 illustrates this fact.

Lemma 3.2. Consider the plant, with input u and output y, 𝑥̇(𝑡)(𝑡)(𝑢(𝑡)),𝑦(𝑡)=𝑥̇(𝑡)(𝑡)(𝑢(𝑡)),=𝐴𝑥+𝐵+𝜉=𝐴𝑥+𝐵+𝜉 (3.5) where 𝑥(𝑡)∈ℝ^𝑛,𝑢(𝑡) and 𝜉∈ℝ^𝑚,𝜉 is an unknown but constant disturbance, and the state-derivative controller 𝑢(𝑡)=−𝐾𝑥̇(𝑡),det(𝐼+𝐵𝐾)≠0. (3.6) Suppose that det(𝐴)≠0 and the equilibrium point 𝑥_𝑒=−𝐴⁻¹𝐵𝜉 of the controlled system (3.5) and (3.6) is globally asymptotically stable. Then, 𝑥(∞) is independent of the controller gain 𝐾 and is given by 𝑥(∞)=lim_𝑡→∞𝑥(𝑡)=−𝐴⁻¹𝐵𝜉. (3.7)

Proof. From (3.5) and (3.6) note that 𝑥̇(𝑡)(𝑡)(𝑥̇(𝑡)),()𝑥̇(𝑡)(𝑡)𝑥̇(𝑡)=()=𝐴𝑥+𝐵−𝐾+𝜉𝐼+𝐵𝐾=𝐴𝑥+𝐵𝜉,𝐼+𝐵𝐾⁻¹𝐴𝑥(𝑡)+(𝐼+𝐵𝐾)⁻¹𝐵𝜉. (3.8) Applying the Laplace transform in (3.8), observe that 𝑠𝑋(𝑠)−𝑥(0)=(𝐼+𝐵𝐾)⁻¹𝐴𝑋(𝑠)+(𝐼+𝐵𝐾)⁻¹𝐵𝜉𝑠⁻¹,𝑋(𝑠)=𝑠𝐼−(𝐼+𝐵𝐾)⁻¹𝐴⁻¹(𝐼+𝐵𝐾)⁻¹𝐵𝜉𝑠⁻¹+𝑥(0). (3.9) Thus, considering that the equilibrium point 𝑥_𝑒 of (3.5) and (3.6) is globally asymptotically stable, from the final value theorem in [1] it follows that 𝑥(∞)=lim_𝑠→0𝑠𝑋(𝑠)=lim_𝑠→0𝑠𝑠𝐼−(𝐼+𝐵𝐾)⁻¹𝐴⁻¹(𝐼+𝐵𝐾)⁻¹𝐵𝜉𝑠⁻¹=−𝐴⁻¹(𝐼+𝐵𝐾)(𝐼+𝐵𝐾)⁻¹𝐵𝜉=−𝐴⁻¹𝐵𝜉. (3.10) This establishes the lemma.

Lemma 3.2 shows that 𝑥(∞) given in (3.7) is independent of the state-derivative matrix 𝐾 defined in (3.6). This fact illustrates the difficulty to attenuate the influence of 𝜉 in 𝑥(∞) in controlled systems with state-derivative feedback, for instance, given by (3.5) and (3.6).

Fortunately, the main application of derivative feedback is in control of vibrations in mechanical systems, where in general the disturbances have a mean value equal to zero.

4. Example

Consider the mass-spring system with damping studied in [24] and shown in Figure 1. In this example, the idea was to illustrate the proposed results in Lemma 3.2. This system is a simple model of a controlled vibration absorber, in the sense of reducing the oscillations of the masses 𝑚₁ and 𝑚₂. The coefficients of elasticity of the springs are 𝑘₁ and 𝑘₂ and 𝑏₁ is the damping coefficient. In this case, the model contains two control inputs, 𝑢₁(𝑡) and 𝑢₂(𝑡), and two unknown but constant disturbances, 𝜉₁ and 𝜉₂. This system is described by the following equations [6]:𝑚₁𝑦̈₁(𝑡)+𝑏₁𝑦̇₁(𝑡)−𝑦̇₂(𝑡)+𝑘₁𝑦₁(𝑡)=𝑢₁(𝑡)+𝜉₁,𝑚₂𝑦̈₂(𝑡)+𝑏₁𝑦̇₂(𝑡)−𝑦̇₁(𝑡)+𝑘₂𝑦₂(𝑡)=𝑢₂(𝑡)+𝜉₂. (4.1) The state-space form of the mechanical system in Figure 1 can be represented by (3.5), considering as state variables 𝑥(𝑡)=[𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)𝑥₄(𝑡)]^𝑇, where 𝑥₁(𝑡)=𝑦₁(𝑡), 𝑥₂(𝑡)=𝑦̇₁(𝑡), 𝑥₃(𝑡)=𝑦₂(𝑡), 𝑥₄(𝑡)=𝑦̇₂(𝑡), 𝑢(𝑡)=[𝑢₁(𝑡)𝑢₂(𝑡)]^𝑇, 𝜉=[𝜉₁𝜉₂]^𝑇, 𝐴=0100−𝑘₁𝑚₁−𝑏₁𝑚₁0𝑏₁𝑚₁0𝑏0001₁𝑚₂−𝑘₂𝑚₂−𝑏₁𝑚₂1,𝐵=00𝑚₁00100𝑚₂. (4.2) Consider the state-derivative feedback control law (3.6). Then, the controlled system (3.5), (3.6), and (4.2) can be described by (3.8).

Figure 1 

Multivariable (MI) mass-spring system with damping.

For the design of a suitable gain 𝐾 using the results from Theorem 1 in [24], first define the following matrices: 𝐴_𝑛=𝐴⁻¹,𝐵_𝑛=−𝐴⁻¹𝐵. (4.3) Then, from (4.2), (4.3) and 𝑚₁=10kg,𝑚₂=30kg,𝑘₁=2.5kN/m,𝑘₂=1.5kN/m, and 𝑏₁=30Ns/m, one has 𝐴_𝑛=,𝐵−0.0120−0.00400.012001.00000000.02000−0.0200−0.0200001.00000_𝑛=0.4000×10⁻³00000.6667×10⁻³.00 (4.4) Note that (𝐴_𝑛,𝐵_𝑛) is controllable.

The idea of Theorem 1 in [24] was the design of state-derivative feedback gains, using a designed state feedback gain.

Thus, consider the pole placement as design technique and the following closed-loop poles for the controlled system (3.8): 𝜆₁=−10,𝜆₂=−15,𝜆_3,4=−2±10𝑖. (4.5) Now, the following closed-loop system is defined [24]: 𝑥̇_𝑛(𝑡)=𝐴_𝑛𝑥(𝑡)+𝐵_𝑛𝑢_𝑛(𝑡),𝑢_𝑛(𝑡)=−𝐾𝑥_𝑛(𝑡). (4.6) From (4.6) it follows that𝑥̇_𝑛(𝑡)=𝐴_𝑛−𝐵_𝑛𝐾𝑥_𝑛(𝑡). (4.7) Then, design the gain 𝐾 such that the system (4.7) has poles equal to 𝜆₁⁻¹,𝜆₂⁻¹,𝜆₃⁻¹, and 𝜆₄⁻¹ [24].

Now, from (4.3) and 𝜆_𝑖≠0, 𝑖=1,2,3,4, note that 𝐴_𝑛−𝐵_𝑛𝐾⁻¹=𝐴⁻¹+𝐴⁻¹𝐵𝐾⁻¹=𝐴⁻¹(𝐼+𝐵𝐾)⁻¹=(𝐼+𝐵𝐾)⁻¹𝐴, (4.8) and from (4.7) and Lemma 1 in [24], 𝜆₁,𝜆₂,𝜆₃, and 𝜆₄ are the eigenvalues of (𝐼+𝐵𝐾)⁻¹𝐴. Therefore, 𝜆₁,𝜆₂,𝜆₃, and 𝜆₄ are also the poles of the controlled system (3.5), (3.6), and (4.2), described in (3.8).

From Theorem 1 in [24], the poles for the new closed-loop system with state feedback (4.7), with 𝐴_𝑛 and 𝐵_𝑛 given in (4.4), are the following:𝜆₁⁻¹=−0.1000,𝜆₂⁻¹=−0.0667,𝜆⁻¹_3,4=−0.0192±0.0962𝑖. (4.9) Thus, with these parameters, one can easily obtain, through the command place of MATLAB, the feedback gain matrix 𝐾₀ below: 𝐾₀=178.9532−6.4647323.354219.8478−79.6370−11.4321152.3204−26.1863, (4.10) such that, for 𝐾=𝐾₀, the poles of (4.7) are equal to 𝜆₁⁻¹,𝜆₂⁻¹,𝜆₃⁻¹, and 𝜆₄⁻¹.

Note that, for 𝐾=0, the poles of the system given in (3.8) are equal to𝜆_1,2=−0.14962±15.6198𝑖,𝜆_3,4=−0.5038±7.1074𝑖. (4.11) Table 1 above shows the values of 𝜉₁,𝜉₂, and 𝐾 in the time intervals, adopted in a digital simulation of the controlled system (3.8) and (4.2).

Table 1 

Disturbances and controller gains of the simulated system.

Figures 2 and 3 display the simulation results of the controlled system (3.5), (3.6), and (4.2), which can be given by (3.8), with the initial condition𝑥(0)=000.10.1 (4.12) and the conditions described in Table 1.

Figure 2 

Transient response of the controlled system (3.8) and (4.2), with 𝑥(0)=[0.100.10]𝑇, disturbances and gains given in Table 1.

Figure 3 

Control inputs of the controlled system (3.8) and (4.2), with 𝑥(0)=[0.100.10]𝑇, disturbances and gains given in Table 1.

For 𝜉=[𝜉₁𝜉₂]^𝑇=[3000]^𝑇, at 𝑡∈[20,40) and 𝑡∈[60,80), from Lemma 3.2, (3.7), and (4.2), one has 𝑥(∞)=lim_𝑡→∞𝑥(𝑡)=−𝐴⁻¹000𝐵𝜉=0.12 (4.13) and for 𝜉=[𝜉₁𝜉₂]^𝑇=[0−300]^𝑇, at 𝑡∈[40,60) and 𝑡∈[80,100) then from Lemma 3.2, (3.7) and (4.2) one has 𝑥(∞)=lim_𝑡→∞𝑥(𝑡)=−𝐴⁻¹000𝐵𝜉=−0.2. (4.14) Note that Figure 2 illustrates these results.

5. Conclusions

The paper presented some conclusive results about the observability, stabilizability, and disturbance rejection of linear time-invariant plants with output equal to the state-derivative vector. There exist many results in the literature about these subjects, for multivariable linear time-invariant systems [23]. However, the authors did not find results for the special case where the plant output is the state-derivative vector. Thus, these results can be useful in the analysis and design of control systems with state-derivative feedback. Future researches in this subject include the analysis and control design of nonlinear state-derivative feedback.

Acknowledgments

The authors gratefully acknowledge the financial support by CAPES, FAPESP, and CNPq from Brazil.