Less Conservative Control Design for Linear Systems with Polytopic Uncertainties via State-Derivative Feedback

The motivation for the use of state-derivative feedback instead of conventional state feedback is due to its easy implementation in some mechanical applications, for example, in vibration control of mechanical systems, where accelerometers have been used to measure the system state. Using linear matrix inequalities LMIs and a parameter-dependent Lyapunov functions PDLF allowed by Finsler’s lemma, a less conservative approach to the controller design via state-derivative feedback, is proposed in this work, with and without decay rate restriction, for continuoustime linear systems subject to polytopic uncertainties. Finally, numerical examples illustrate the efficiency of the proposed method.


Introduction
In studies of classical control theory, it is well known that the use of state-derivative feedback u t −K d ẋ t can be very useful, and in some cases, essential and advantageous for achieving desired performance in dynamic systems 1 .The interest to study the statederivative feedback comes from the fact that in systems using accelerometers, it is easier to get the state-derivatives signals than the states signals.Using the acceleration signal, the velocity can be obtained with good accuracy.However, it is more difficult to obtain the displacement 2 .Thus, the signals used in feedback are: velocity and acceleration.Respectively, these are the derivatives of position and velocity that can represent the system states.Due to its simple structure and low operating cost, accelerometers have been used in industry for solving various engineering problems.For example, they can give some applications: in vibration control of suspension bridges cables 3 , in vibration control of components for aircraft Notation.The following notations are used in the text: T indicates transposition of a vector or matrix; −T indicates the inverse of the transposed matrix; * indicates transposed blocks in a symmetric matrix; diag •, •, . . ., • denotes a block diagonal matrix.

State-Derivative Feedback Control in Linear Systems with
Polytopic Uncertainties Consider a linear system with time-invariant uncertainties, described as the state space variables as in 2.1 ẋ t A ξ x t B ξ u t , x t 0 x 0 , 2.1 where A ξ ∈ R n×n and B ξ ∈ R n×m are matrices that represent the dynamics of the uncertain system, x t ∈ R n is the state vector, and u t ∈ R m is the control input vector.
The A, B ξ matrices are represented by the convex combination of vertices, described by 18 where r is given by r 2 φ , and φ is the number of uncertain components of the A, B ξ matrices, and A, B i represent the vertices of the polytope.The ξ i parameters are unknown real constants, belonging to a simplex set C, that is, satisfying the relation: Assuming that the system 2.1 has no null eigenvalues det A ξ / 0 2 , the objective is to find a constant matrix K d ∈ R m×n , such that, when feedback into the system 2.1 with the control input the closed-loop system given by 2.1 and 2.4 is asymptotically stable and the matrix I B ξ K d is invertible det I B ξ K d / 0 .Thus, the closed-loop system can be represented by which is equivalent to expression 2.5 .

Mathematical Problems in Engineering
The controller design is accomplished using the concept of asymptotic stability by analyzing the existence of a parametric Lyapunov function as in 2.7 , for all x t / 0 27 : x t T P ξ x t > 0, P ξ P ξ T ∈ R n×n , V x t , ξ ẋ t T P ξ x t x t T P ξ ẋ t < 0, 2.7 where P ξ is given by In the LMIs modelling, Lemma 2.1 is used.
Then, the following conditions are equivalent: where ρ and Q are called extra variables (or multipliers).
Lemma 2.1 can be used to study a set of matrix inequalities without an explicit multiplication between the matrices of the system 2.6 with the matrices P ξ of the Lyapunov function.This Lemma is widely used in many LMI-based control applications, so as to eliminate variables, decouple matrices, or reduce the number of LMIs in control design 33, 34 .At first, we define the following vectors and matrices: where X is any nonsingular matrix of appropriate dimensions.Using these definitions, the closed-loop system 2.6 and the stability condition 2.7 can be rewritten respectively as During the demonstration of the theorems proposed below, Lemma 2.2 is also used.
where i 1, 2, . . ., r, then, the system 2.6 is stabilizable, and a controller that solves the problem can be found as in Proof.Suppose 2.11 is feasible.

2.13
From 2.2 and 2.8 , 2.14 follows Left-multiplying 2.14 by diag X, X and right-multiplying by diag X T , X T , 2.15 follows

2.15
Multiplying 2.15 by −1 and left-and right-multiplying by 0 I I 0 , 2.16 follows According to Lemma 2.2, it follows that X I B ξ K d is full rank, so I B ξ K d and X T are full rank, that is, invertible.

Mathematical Problems in Engineering
Note that 2.15 can be rewritten as Now, putting X and X T in evidence, 2.19 follows Considering 2.9 and Lemma 2.1, From this it follows that if the LMI 2.11 is feasible, then there exists a symmetric parameter-dependent matrix P ξ > 0 satisfying the conditions of Lyapunov 2.7 for the system 2.6 .Thus, system 2.6 , considering the gain 2.12 , is asymptotically stable.
The stability of the system is not always enough, because there are projects that have performance constraints.A very important performance index is the restriction of decay rate, which is responsible for the rapid response transient period of the system 18 .

Decay Rate in State-Derivative Feedback Control with Polytopic Uncertainties
Consider a candidate Lyapunov function of the type V x t , ξ x t T P ξ x t > 0, with V x t , ξ < 0 for all x t / 0. The decay rate γ > 0 is obtained if the condition is satisfied for every trajectory x t of the system 2.6 , t ≥ 0 18 .Figure 1 shows the region γ, where λ are the eigenvalues of 2.6 .Defining the following vectors and matrices

2.21
the stability condition with decay rate restriction 2.20 can be rewritten as 2.10 .

Im (λ)
Re (λ) Knowing this, sufficient conditions for the stability of system 2.6 with decay rate γ > 0 are proposed in Theorem 2.4.
where i 1, 2, . . ., r, then, the system 2.6 is stabilizable with decay rate γ, and a controller that solves the problem can be found by 2.12 .
Proof.The proof follows similar steps from Theorem 2.3 proof, considering 2.21 .

Extended Results of Finsler's Lemma Applied to State-Derivative Feedback Control
In the next theorems, a scalar μ > 0 is involved in the set of LMIs.It is possible to obtain less conservative results than the theorems presented above with an appropriate choice of μ.The proposed Theorem 3.1 is an extension of Theorem 2.3.
Theorem 3.1.For a given arbitrary scalar μ > 0, if there exist symmetric matrices where i 1, 2, . . ., r, then, the system 2.6 is stabilizable, and a controller that solves the problem can be found by 2.12 .
Proof.The proof follows similar steps from Theorem 2.3 proof, considering The proposed Theorem 3.2 is an extension of Theorem 2.4.
Theorem 3.2.Given a real constant γ > 0 and an arbitrary scalar μ > 0, if there exist symmetric matrices Q i ∈ R n×n , matrices G ∈ R m×n , and Y ∈ R n×n , satisfying the LMI where i 1, 2, . . ., r, then, the system 2.6 is stabilizable with decay rate γ, and a controller that solves the problem can be found by 2.12 .
Proof.The proof follows similar steps from Theorem 2.3 proof, considering 2.21 and 3.2 .
Note 1.In Theorems 3.1 and 3.2 the scalar μ > 0 must be obtained through a one-dimensional search.
The tuning parameter μ was added in the LMIs 3.1 and 3.3 in an attempt to obtain less conservative stability conditions than Theorems 2.3 and 2.4, respectively.This procedure of adding scalar in LMIs has been widely explored in literature.For example, in 30, 31, 33, 35 , the authors can improve the LMI stability region by simply changing the parameter values.
The efficiency of the proposed methodology can be found in the solution of the examples discussed below.

Examples-Digital Simulations
Consider the vibration control system illustrated in Figure 2 2 .
The system's dynamics can be represented by the uncertain system 2.1 , considering with the state vector given by x t x 1 t x 2 t ẋ1 t ẋ2 t T , where, x 1 t and x 2 t are the vertical displacements of the masses m 1 and m 2 , respectively, and ẋ1 t and ẋ2 t are their Accelerometer ẍ2 (t)) x 1 (t)  1.
Assuming that the damping coefficient b 1 is uncertain and belongs to the interval 0 ≤ b 1 ≤ 70 Ns/m i.e., the damper may break after some time of use, b 1 0 .The objective of the control system is to reduce the vibration of the mass m 1 through of the controlled vibration of the mass m 2 , with the control signal u t .
Using this information, the following vertices of the polytope are obtained:

4.4
For the controllers design, MatLab software and the solver "LMILab" 19 were used.For this example, it was considered the following case.

Stability
Using the LMI 2.11 from Theorem 2.3, the controller in 4.5 was obtained using 2.12 K d 10 4 × −4.3659 3.6076 0.0292 0.0075 .4.5 The controlled system response to the initial condition of simulation x 0 0.05 0.05 0.2 0.2 T with damper b 1 working and after the failure b 1 0 can be seen in Figure 3. Referring to Figure 3, note that the behavior of the controlled system almost does not change, even after breaking the damper b 1 .Thus, the designed controller was able to stabilize the system even after the occurrence of a structural failure.
Figure 4 shows the control signal effort considering controller 4.5 in control input 2.4 .Now, using the LMI 3.1 from Theorem 3.1 with μ 20, the controller in 4.6 was obtained using 2.12 K d 10 4 × −1.2108 1.0586 0.0041 0.0011 .

4.6
Figure 6 shows the control signal effort considering controller 4.6 in control input 2.4 .
In this example, with a one-dimensional search in μ, controller 4.6 found by Theorem 3.1 obtained a better performance considering the control signal input Figure 6 than controller 4.5 found by Theorem 2.3 Figure 4 .However, there are no theoretical guarantees about the influence of μ in the performance of the closed-loop system.What we can guarantee is that there are systems that can be stabilized with Theorem 3.1 and cannot be with Theorem 2.3.This is verified later in Section 4.1.
Still looking at Figures 3 and 5, the duration of the transient period of the controlled system in both cases is around 3.5 seconds; that is, the tuning parameter μ has had limited influence in this regard.Whereas the system must have a lower transient, because in some cases this time may be too long, new projects can be done by adding the decay rate restriction.Using Theorems 2.4 and 3.2, a shorter transient period response can be found.

Stability and Decay Rate
Using the LMI 2.22 from Theorem 2.4 with γ 0.99 maximum obtained for this example , the controller in 4.7 was obtained using 2.12 K d 10 3 × −5.0893 4.5906 0.0577 0.0038 .

4.7
The response of the controlled system with decay rate γ 0.99 with damper b 1 working and with damper b 1 broken structural failure can be seen in Figure 7.By examining Figure 7, it can be observed that the behavior of the controlled system is almost not affected, even after the occurrence of a failure in the damper b 1 .Thus, the designed controller was able to secure the stability of the system and even reduce the settling time at around 1.5 s.In Table 2, it is clear that the eigenvalues of the local models have real part lesser than γ 0.99.Therefore, the decay rate specification of the project has been met.There are four eigenvalues for each vertex.Note 2. After lots of tests, through numerical examples, it was not possible to obtain values of γ > 0.99 using the LMI 2.22 from Theorem 2.4.However, this is overcome by using Theorem 3.2.Note that with the addition of the tuning parameter μ in LMIs, it was possible to design controller with decay rate greater than 0.99 γ > 0.99 , which hitherto was not possible.Therefore, the use of the parameter μ on designs that consider the stabilization of the system with decay rate can establish influences in transient behavior of the controlled system, given that the greater the decay rate, the smaller the transient and, moreover, the control effort increases.Observing Figure 9, it can be seen that the controlled system, with or without fault, has settling time under 0.4 s.Thus, the controller 4.8 was able to improve system performance comparing to responses of the controllers 4.5 , 4.6 , and 4.7 .The response to 4.8 is approximately 12 times faster than the response with 4.5 and 4.6 and 7 times faster than 4.7 .
In Table 3, it is clear that the eigenvalues of the local models have real part lesser than γ 10.Therefore, the decay rate specification of the project has been met.There are four eigenvalues for each vertex.Note 3.After lots of tests, the maximum value found for this example was γ 162.88 with μ 0.00001.Considering these parameters, the controlled system has a settling time of approximately 0.04 s.Consequently, there is a considerable increase of the control effort approximately max |u t | 1 × 10 10 N .
Figure 10 shows the control signal effort considering the controller 4.8 in the control input 2.4 .In the next example it is illustrated that the stability conditions considering μ in the design are less conservative.

Example 2-Linear Systems with Polytopic Uncertainties
Consider an uncertain linear system with the following polytope vertices: 4.9

Stability
The objective is to verify the feasibility points of Theorems 2. Considering the same uncertain linear system, Figure 12 shows the totality of controllers that could be found for various values of μ, comprising μ ∈ 0.05, 2.5 .Analysing Figure 12, it can be observed that for small values of μ the totality of feasible designs is greater.

Stability and Decay Rate
Now, using Theorems 2.4 and 3.2 and considering −10 ≤ ζ 1 ≤ 30 and 0.4 ≤ ζ 2 ≤ 5.6, the feasibility region with γ 0.99 is shown in Figure 13.Intentionally, the choice of γ 0.99 is due to the reason mentioned in Note 2.
Gradually increasing the value of the parameter γ, Theorem 3.2 can still find the controllers for stabilizing uncertain linear system, while designs without the parameter μ are no longer feasible.This is shown in Figure 14.
Considering the same uncertain linear system with −20 ≤ ζ 1 ≤ 30 and −1.6 ≤ ζ 2 ≤ 2.4, Figure 15 shows the totality of controllers that could be found for various values of γ and μ, comprising γ ∈ 0, 5 and μ ∈ 0.01, 1.2 .Analysing Figure 15, it can be observed that for a large value of γ, it is required a small value of μ for feasibility.
In this example, note that the values of μ are different for each design situation.After a one-dimensional search and visually considering Figures 12 and 15, these were the best values for the uncertain system.However, there is no theoretical evidence that these are the optimal values of μ.The next example illustrates a comparison of the stability criteria proposed in this paper with those proposed in 25 and in 31 .In 25 , a CQLF is used to ensure the stability of the system with and without decay rate γ > 0, and in 31 , a PDLF is used for the same purposes.

Example 3-Comparison between LMI-Based Techniques via State-Derivative Feedback Control
Consider an uncertain linear system with the following polytope vertices:

Stability
Considering −30 ≤ ζ 1 ≤ 102 and 60 ≤ ζ 2 ≤ 111, the region of feasibility considering only the stability of the system can be seen in Figure 16.Note from Figure 16 that using only Theorem 2.3 from this work, which is more conservative than Theorem 3.1 of the same work, it has been possible to obtain less conservative conditions than those presented in 25 and in 31 .
Considering the restriction of decay rate γ 2, in this example it was used Theorem 3.2, knowing that Theorem 2.4 has no feasible answers for this value of γ.From Figure 17, note that the stability conditions proposed in this paper are less conservative than those presented in 25 and in 31 .Theorems 2.3 and 2.4 proposed in 31 have not been  used in both comparisons given that they are more conservative than Theorems 3.1 and 3.2 of the same work.

Conclusions
Sufficient conditions based on LMI directed for the state-derivative feedback controllers design, with and without decay rate restriction of uncertain linear systems using PDLFs, have been proposed.It is seen that the addition of the tuning parameter in LMI achieved better results, especially when considering the decay rate in the designs.The technique  is particularly interesting for applications in systems using accelerometers as sensors.The methodology addressed decouples the system matrices from the Lyapunov function matrices, thus improving the solution of problems.Comparisons between LMIs techniques using CQLFs and PDLFs functions were presented and discussed, and the results illustrate a less conservative search by controllers using the method proposed in this paper.

Figure 8 8 Displacementx 2 Figure 7 :
Figure8shows the control signal effort considering controller 4.7 in control input 2.4 .Using the LMI 3.3 from Theorem 3.2 with μ 0.01 and γ 10, the controller in 4.8 was obtained using 2.12

Figure 12 :
Figure 12: Parameters for which the controllers are found μ × Feasibility .

Lemma 2.2. For
every nonsymmetric matrix M M / M T , if M M T < 0, then M is invertible.

Theorem 2.3. If
there exist symmetric matrices Q i ∈ R n×n , matrices G ∈ R m×n , and Y ∈ R n×n , satisfying the LMI and Y X −1 in 2.11 , multiplying each element by ξ i and summing for all i 1, 2, . . ., r, 2.13 follows

Table 1 :
Parameters of mechanical system.
velocities.Parameters k 1 and k 2 are elastic constants and b 1 and b 2 damping constants.The respective values for the system parameters are shown in Table

Table 2 :
Eigenvalue position for system 4.1 with 4.7 state-derivative feedback gain.

Table 3 :
Eigenvalue position for system 4.1 with 4.8 state-derivative feedback controller.