Robust State-Derivative Feedback LMI-Based Designs for Linear Descriptor Systems

Abstract

Techniques for stabilization of linear descriptor systems by state-derivative feedback are proposed. The methods are based on Linear Matrix Inequalities (LMIs) and assume that the plant is a controllable system with poles different from zero. They can include design constraints such as: decay rate, bounds on output peak and bounds on the state-derivative feedback matrix 𝐾, and can be applied in a class of uncertain systems subject to structural failures. These designs consider a broader class of plants than the related results available in the literature. The LMI can be efficiently solved using convex programming techniques. Numerical examples illustrate the efficiency of the proposed methods.

1. Introduction

The Linear Matrix Inequalities (LMIs) formulation has emerged recently as a useful tool for solving a great number of practical control problems [1–10]. Furthermore, LMI can be solved with polynomial convergence time, by convex optimization algorithms [1, 11–13].

Recently, LMI has been used for the study of descriptor systems [14–17]. Descriptor systems can be found in various applications, for instance, in electrical systems, or in robotics [18]. The proportional and derivative feedback (𝑢=𝐿𝑥(𝑡)−𝐾𝑥̇(𝑡), where 𝑥(𝑡) is the plant state vector) has been studied by many authors to design controllers in the following problems: stabilization and regularizability of linear descriptor systems [19, 20], feedback control of singular systems [21], nonlinear control with exact feedback linearization [22], ℋ_∞-control of continuous-time systems with state delay [23], and design of PD observers [24]. In [18, 25] some properties of this type of feedback and its applications to pole placement were presented.

There exist few researches using only derivative feedback (𝑢=−𝐾𝑥̇(𝑡)). In some practical problems the state-derivative signals are easier to obtain than the state signals, for instance, in the following applications: suppression of vibration in mechanical systems [26], control of car wheel suspension systems [27], vibration control of bridge cables [28], and vibration control of landing gear components [29]. The main sensors used in these problems 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 [26]. Defining the velocities and displacement as the state variables, then one has available for feedback only the state-derivative signals. Procedures for solving the pole-placement problem for linear systems using state-derivative feedback were proposed in [26, 30, 31]. In [28, 32] a Linear Quadratic Regulator (LQR) controller design scheme for standard state space systems was presented. The results were obtained in Reciprocal State Space (RSS) framework. Robust state-derivative feedback LMI-based designs for linear time-invariant systems were recently proposed in [33, 34]. These results considered only standard linear systems, and they can be applied to uncertain systems, with or without, structural failures.

Structural failures appear naturally in systems, for physical wear of the equipment, or for short circuit of electronic components. Recent researches on structural failures (or faults), have been presented in LMI framework [35–38].

In this paper, we will show that it is possible to extend the presented results in [33], for applications in a class of descriptor systems, subject to structural failures in the plant. The procedure can include some specifications: decay rate, bounds on output peak and bounds on the state-derivative feedback matrix 𝐾, which can make easier the practical implementation of the controllers. These methods allow new specifications, and also to consider a broader class of plants that the related results are available in the literature [19, 25, 31, 39]. Two examples illustrate the efficiency of the proposed method.

2. Statement of the Problem

Consider a controllable linear descriptor system described by

𝐸𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), (2.1) where 𝑥(𝑡)∈ℝ^𝑛,  𝑢(𝑡)∈ℝ^𝑚,  𝐸∈ℝ^𝑛×𝑛,  𝐴∈ℝ^𝑛×𝑛, and 𝐵∈ℝ^𝑛×𝑚. It is known that the stability problem for descriptor systems is more complicated than for standard systems, because it requires considering not only stability, but also regularity [15, 25]. In the next sections, LMI conditions for asymptotic stability of descriptor system (2.1) using state-derivative feedback, are proposed. The problem is defined as follows.

Problem 1. Find a constant matrix 𝐾∈ℝ^𝑚×𝑛, such that the following conditions hold: (1)(𝐸+𝐵𝐾) has a full rank; (2)the closed-loop system (2.1) with the state-derivative feedback control 𝑢(𝑡)=−𝐾𝑥̇(𝑡), (2.2) is regular and asymptotically stable (in this work, a descriptor system is regular if it has uniqueness in the solutions and avoid impulsive responses).

Remark 2.1. In [25, 39] the authors assure that (𝐸+𝐵𝐾) has a full rank (nonsingular matrix) only if the following equation holds: rank[𝐸,𝐵]=𝑛. (2.3)
Unfortunately, there exist several practical problems that not satisfy (2.3). In that way, the input control (2.2) can only be applied in descriptor systems (2.1), when (2.3) holds. Some authors have been using the state-derivative and state feedback (𝑢=𝐿𝑥(𝑡)−𝐾𝑥̇(𝑡)) to solve (2.1), when (2.3) does not hold [18, 20]. However, usually these designs are more complex than the design procedures with only state or state-derivative feedback.

Assuming that (𝐸+𝐵𝐾) has a full rank, then from (2.2) it follows that (2.1) can be rewrite such as a standard linear system, given by

𝐸𝑥̇(𝑡)=𝐴𝑥(𝑡)−𝐵𝐾𝑥̇(𝑡)⇔𝑥̇(𝑡)=(𝐸+𝐵𝐾)⁻¹𝐴𝑥(𝑡). (2.4)

3. LMI-Based Stability Conditions for State-Derivative Feedback

Necessary and sufficient conditions for asymptotic stability of standard linear system (2.4) are proposed in the next theorems.

Theorem 3.1. Assuming that (2.3) holds, the necessary and sufficient condition for the solution of Problem 1 is the existence of matrices 𝑄=𝑄^′,  𝑄∈ℝ^𝑛×𝑛 and 𝑌∈ℝ^𝑚×𝑛, such that, 𝐴𝑄𝐸^′+𝐸𝑄𝐴^′+𝐵𝑌𝐴^′+𝐴𝑌^′𝐵^′<0,𝑄>0. (3.1)(3.2) Furthermore, when (3.1) and (3.2) hold, then a state-derivative feedback matrix that solves Problem 1 can be given by 𝐾=𝑌𝑄⁻¹. (3.3)

Proof. Observe that for any nonsymmetric matrix 𝑀 (𝑀≠𝑀^′), 𝑀∈ℝ^𝑛×𝑛, if 𝑀+𝑀^′<0, then 𝑀 has a full rank. Now, defining 𝑄=𝑃⁻¹ and 𝑌=𝐾𝑄, the following equations are equivalents: 𝐴𝑄𝐸^′+𝐸𝑄𝐴^′+𝐵𝑌𝐴^′+𝐴𝑌^′𝐵^′=𝐴𝑄(𝐸+𝐵𝐾)^′+(𝐸+𝐵𝐾)𝑄𝐴^′<0⇔𝑃(𝐸+𝐵𝐾)⁻¹𝐴+𝐴^′[(𝐸+𝐵𝐾)⁻¹]^′𝑃<0, (3.4)(3.5) From (3.4) one has the matrix (𝐸+𝐵𝐾)𝑄𝐴^′ has full rank, and so, (𝐸+𝐵𝐾) also has a full rank, as required in Problem 1, and (3.5) was obtained after premultiplying by 𝑃(𝐸+𝐵𝐾)⁻¹ and posmultiplying by [(𝐸+𝐵𝐾)⁻¹]^′𝑃 in both sides of (3.4).
System (2.4) is globally asymptotically stable only if there exists 𝑃=𝑃^′>0 (that is equivalent to 𝑄=𝑄^′=𝑃⁻¹>0) such that (3.4) or (3.5) holds.

Remark 3.2. Note that from (3.4) it follows that matrix 𝐴 must have a full rank, and so, all its eigenvalues are different from zero. This condition was also considered in other papers [26, 28, 33] for linear systems.

Equations (3.1) and (3.2) are LMI. When (3.1) and (3.2) are feasible, they can be easily solved using available software, such as LMISol [40], that is a free software, or MATLAB [11]. The algorithms have polynomial time convergence.

Usually, only the stability of the control systems is insufficient to obtain a suitable performance. In the design of control systems, the specification of the decay rate can also be very useful.

3.1. Decay Rate in State-Derivative Feedback

Consider, for instance, the controlled system (2.4). According to [1], the decay rate is defined as the largest real constant 𝛾, 𝛾>0, such that,

lim_𝑡→∞𝑒^𝛾𝑡‖𝑥(𝑡)‖=0 (3.6) holds, for all trajectories 𝑥(𝑡), 𝑡≥0.

Theorem 3.3. Assuming that (2.3) holds, the closed-loop system given by (2.4), in Problem 1, has decay rate greater or equal to 𝛾 if there exist matrices 𝑄=𝑄^′ and 𝑌, where 𝑄∈ℝ^𝑛×𝑛 and 𝑌∈ℝ^𝑚×𝑛, such that: 𝐴𝑄𝐸^′+𝐸𝑄𝐴^′+𝐵𝑌𝐴^′+𝐴𝑌^′𝐵^′𝐸𝑄+𝐵𝑌𝑄𝐸^′+𝑌^′𝐵^′−𝑄2𝛾<0,𝑄>0. (3.7)(3.8) Furthermore, when (3.7) and (3.8) hold, then a state-derivative feedback matrix can be given by: 𝐾=𝑌𝑄⁻¹. (3.9)

Proof. Stability corresponds to positive decay rate, 𝛾>0. One can use the quadratic Lyapunov function 𝑉(𝑥(𝑡))=𝑥^′(𝑡)𝑃𝑥(𝑡) to impose a lower bound on the decay rate with 𝑉̇(𝑥(𝑡))<−2𝛾𝑉(𝑥(𝑡)), as described in [1]. Note that, from (2.4), 𝑉̇(𝑥(𝑡))=𝑥̇^′(𝑡)𝑃𝑥(𝑡)+𝑥^′(𝑡)𝑃𝑥̇(𝑡)=𝑥^′(𝑡)𝐴^′[(𝐸+𝐵𝐾)⁻¹]^′𝑃𝑥(𝑡)+𝑥^′(𝑡)𝑃(𝐸+𝐵𝐾)⁻¹𝐴𝑥(𝑡). (3.10) Then, from 𝑉̇(𝑥(𝑡))<−2𝛾𝑉(𝑥(𝑡)) it follows that, 𝑥^′(𝑡)𝐴^′[(𝐸+𝐵𝐾)⁻¹]^′𝑃𝑥(𝑡)+𝑥^′(𝑡)𝑃(𝐸+𝐵𝐾)⁻¹𝐴𝑥(𝑡)<−2𝛾𝑥^′(𝑡)𝑃𝑥(𝑡), (3.11) or 𝐴^′(𝑃⁻¹𝐸^′+𝑃⁻¹𝐾^′𝐵^′)⁻¹+(𝐸𝑃⁻¹+𝐵𝐾𝑃⁻¹)⁻¹𝐴<−2𝛾𝑃. (3.12) After premultiplying by (𝐸𝑃⁻¹+𝐵𝐾𝑃⁻¹) and posmultiplying by (𝑃⁻¹𝐸^′+𝑃⁻¹𝐾^′𝐵^′) in both sides of (3.12), observe that (3.12) holds if and only if (𝐸𝑃⁻¹+𝐵𝐾𝑃⁻¹)𝐴^′+𝐴(𝑃⁻¹𝐸^′+𝑃⁻¹𝐾^′𝐵^′)<(𝐸𝑃⁻¹+𝐵𝐾𝑃⁻¹)(−2𝛾𝑃)(𝑃⁻¹𝐸^′+𝑃⁻¹𝐾^′𝐵^′) (3.13) and so −(𝐸𝑃⁻¹+𝐵𝐾𝑃⁻¹)𝐴^′−𝐴(𝑃⁻¹𝐸^′+𝑃⁻¹𝐾^′𝐵^′)−(−1)(𝐸𝑃⁻¹+𝐵𝐾𝑃⁻¹)(2𝛾𝑃)(−1)(𝑃⁻¹𝐸^′+𝑃⁻¹𝐾^′𝐵^′)>0. (3.14) Now, using the Schur complement [1], the equation above is equivalent to −(𝐸𝑃⁻¹+𝐵𝐾𝑃⁻¹)𝐴^′−𝐴(𝑃⁻¹𝐸^′+𝑃⁻¹𝐾^′𝐵^′)−(𝐸𝑃⁻¹+𝐵𝐾𝑃⁻¹)−(𝑃⁻¹𝐸^′+𝑃⁻¹𝐾^′𝐵^′)𝑃⁻¹(2𝛾)>0. (3.15) Therefore, defining 𝑄=𝑃⁻¹ and 𝑌=𝐾𝑃⁻¹, then it follows the expression (3.7). If 𝑃>0 then 𝑄>0, as specified in (3.8). So, when (3.7) and (3.8) hold, a state-derivative feedback matrix 𝐾 is given by (3.9).

The next section shows that it is possible to extend the presented results, for the case where there exist polytopic uncertainties or structural failures in the plant. A fault-tolerant design is proposed.

4. Robust Stability Condition for State-Derivative Feedback

In this work, structural failure is defined as a permanent interruption of the system's ability to perform a required function under specified operating conditions [41]. Systems subject to structural failures can be described by uncertain polytopic systems.

Consider the linear time-invariant uncertain polytopic descriptor system, with or without structural failures, described as convex combinations of the polytope vertices:

^𝑟𝑒_𝑖=1𝑒_𝑖𝐸_𝑖𝑥̇(𝑡)=^𝑟𝑎_𝑗=1𝑎_𝑗𝐴_𝑗𝑥(𝑡)+^𝑟𝑏_𝑘=1𝑏_𝑘𝐵_𝑘𝑒𝑢(𝑡),_𝑖≥0,𝑖=1,…,𝑟_𝑒,^𝑟𝑒_𝑖=1𝑒_𝑖𝑎=1,_𝑗≥0,𝑗=1,…,𝑟_𝑎,^𝑟𝑎_𝑖=1𝑎_𝑗𝑏=1,_𝑘≥0,𝑘=1,…,𝑟_𝑏,^𝑟𝑏_𝑗=1𝑏_𝑘=1, (4.1)(4.2) where 𝑟_𝑒,𝑟_𝑎, and 𝑟_𝑏 are the numbers of polytope vertices of 𝐸, 𝐴, and 𝐵, respectively. In (4.2), 𝑒_𝑖,𝑎_𝑗, and 𝑏_𝑘, are constant and unknown real numbers for all index 𝑖,𝑗,𝑘. The next theorem solves Problem 1, replacing system (2.1) by the uncertain system (4.1).

Theorem 4.1. A sufficient condition for the solution of Problem 1 for the uncertain system (4.1) is the existence of matrices 𝑄=𝑄^′ and 𝑌, where 𝑄∈ℝ^𝑛×𝑛 and 𝑌∈ℝ^𝑚×𝑛, such that, 𝐴_𝑗𝑄𝐸^′_𝑖+𝐸_𝑖𝑄𝐴^′_𝑗+𝐵_𝑘𝑌𝐴^′_𝑗+𝐴_𝑗𝑌^′𝐵^′_𝑘<0,𝑄>0, (4.3)(4.4) where 𝑖=1,2,…,𝑟_𝑒, 𝑗=1,2,…,𝑟_𝑎, and 𝑘=1,2,…,𝑟_𝑏. Furthermore, when (4.3) and (4.4) hold, then a state-derivative feedback matrix can be given by, 𝐾=𝑌𝑄⁻¹. (4.5)

Proof. From (4.2) and (4.3) it follows that ^𝑟𝑒_𝑖=1𝑒_𝑖^𝑟𝑎_𝑗=1𝑎_𝑗^𝑟𝑏_𝑘=1𝑏_𝑘[𝐴_𝑗𝑄𝐸^′_𝑖+𝐸_𝑖𝑄𝐴^′_𝑗+𝐵_𝑘𝑌𝐴^′_𝑗+𝐴_𝑗𝑌^′𝐵^′_𝑘]=(^𝑟𝑎_𝑗=1𝑎_𝑗𝐴_𝑗)𝑄(^𝑟𝑒_𝑖=1𝑒_𝑖𝐸^′_𝑖)+(^𝑟𝑒_𝑖=1𝑒_𝑖𝐸_𝑖)𝑄(^𝑟𝑎_𝑗=1𝑎_𝑗𝐴^′_𝑗)+(^𝑟𝑏_𝑘=1𝑏_𝑘𝐵_𝑘)𝑌(^𝑟𝑎_𝑗=1𝑎_𝑗𝐴^′_𝑗)+(^𝑟𝑎_𝑗=1𝑎_𝑗𝐴_𝑗)𝑌^′(^𝑟𝑏_𝑘=1𝑏_𝑘𝐵^′_𝑘)<0. (4.6) Therefore, condition (3.1) of Theorem 3.1 holds for the uncertain system (4.1), where 𝐸=𝑒₁𝐸₁+⋯+𝑒_𝑟𝑒𝐸_𝑟𝑒,𝐴=𝑎₁𝐴₁+⋯+𝑎_𝑟𝑎𝐴_𝑟𝑎, and 𝐵=𝑏₁𝐵₁+⋯+𝑏_𝑟𝑏𝐵_𝑟𝑏. Now, conditions (4.4) and (4.5) are equivalent to conditions (3.2) and (3.3). Finally, from Theorem 3.1, the existence of matrices 𝑄=𝑄^′ and 𝑌 such that (4.3) and (4.4) hold is a sufficient condition for the solution of Problem 1.

Theorem 4.2. A sufficient condition for the decay rate of the robust closed-loop system given by (2.2) and (4.1) to be greater or equal to 𝛾 is the existence of matrices 𝑄=𝑄^′ and 𝑌, 𝑄∈ℝ^𝑛×𝑛, 𝑌∈ℝ^𝑚×𝑛, such that: 𝐴_𝑗𝑄𝐸^′_𝑖+𝐸_𝑖𝑄𝐴^′_𝑗+𝐵_𝑘𝑌𝐴^′_𝑗+𝐴_𝑗𝑌^′𝐵^′_𝑘𝐸_𝑖𝑄+𝐵_𝑘𝑌𝑄𝐸^′_𝑖+𝑌^′𝐵^′_𝑘−𝑄2𝛾<0,∀𝑖,𝑗,𝑄>0. (4.7) Furthermore, when (4.7) hold, then a robust state-derivative feedback matrix can be given by 𝐾=𝑌𝑄⁻¹. (4.8)

Proof. It follows directly from the proofs of Theorems 3.3 and 4.1.

Due to limitations imposed in the practical applications of control systems, many times it should be considered output constraints in the design.

5. Bounds on Output Peak

Consider that the output of the system (2.1) is given by

𝑦(𝑡)=𝐶𝑥(𝑡), (5.1) where 𝑦(𝑡)∈ℝ^𝑝 and 𝐶∈ℝ^𝑝×𝑛. Assume that the initial condition of (2.1) and (5.1) is 𝑥(0). If the feedback system (2.1), (2.2), and (5.1) is asymptotically stable, one can specify bounds on output peak as described in:

max‖𝑦(𝑡)‖₂=max𝑦^′(𝑡)𝑦(𝑡)<𝜉₀, (5.2) for 𝑡≥0, where 𝜉₀ is a known positive constant. From [1], (5.2) is satisfied when the following LMI holds:

1𝑥(0)^′𝑥(0)𝑄>0,𝑄𝑄𝐶^′𝐶𝑄𝜉²₀𝐼>0, (5.3) and the LMI that guarantees stability (Theorem 3.1 or Theorem 4.1), or stability and decay rate (Theorem 3.3 or Theorem 4.2).

An interesting method for specification of bounds on the state-derivative feedback matrix 𝐾 was recently proposed in [33]. The result is presented below.

Lemma 5.1. Given a constant 𝜇₀>0, then the specification of bounds on the state-derivative feedback matrix 𝐾 can be described finding the minimum of 𝛽, 𝛽>0, such that 𝐾𝐾^′<𝛽𝐼/𝜇²₀. The optimal value of 𝛽 can be obtained by the solution of the following optimization problem: 𝑌min𝛽𝑠.𝑡.𝛽𝐼𝑌_′𝐼>0,𝑄>𝜇₀𝐼,(SetofLMI), (5.4) where the Set of LMI can be equal to (3.1), (3.2) or (3.7), (3.8) or (4.3), (4.4) or (4.7), with or without the LMI (5.3).

Proof. See [33].

In the following section, Example 6.1 illustrates the efficiency of this optimization procedure that can reduce the practical difficulties in the implementation of the controllers.

6. Examples

The effectiveness of the proposed LMI designs is demonstrated by simulation results.

Example 6.1. A simple electrical circuit, can be represented by the linear descriptor system below [25]: 01𝑥̇00𝑥̇¹(𝑡)₂=𝑥(𝑡)1001𝑥¹(𝑡)₂+0(𝑡)1𝑢(𝑡), (6.1) where 𝑥₁ is the current and the 𝑥₂ is the potential of the capacitor.
Suppose the output of the system is given by 𝑦(𝑡)=𝑥₁. So it is a Single-Input/Single-Output (SISO) system, with 𝑛=2, 𝑚=1 and 𝑝=1. Consider as specification an output peak bound 𝜉₀=10 and an initial condition equal to 𝑥(0)=[10]^′. Then, using the package “LMI control toolbox” from MATLAB [11] to solve the LMI (3.1) and (3.2) from Theorem 3.1, and (5.3), one feasible solution was obtained
,.𝑄=59.366−16.491−16.49198.944𝑌=−98.944−49.472 (6.2) A state-derivative feedback matrix was calculated using (3.3) 𝐾=−1.8932−0.81553. (6.3)
Note that, as discussed before, the obtained solution 𝐾 is such that det(𝐸+𝐵𝐾)≠0 (it is equal to 1.8932).
For the initial condition 𝑥(0) given above, the simulation results of the controlled system are presented in Figure 1. From Figure 1, the settling time of the controlled system is approximately 25 seconds and max𝑦^′(𝑡)𝑦(𝑡) is equal to 1<𝜉₀=10. The specification for the controlled system was satisfied using the designed controller. Note by Figure 1 that only the stability of the controlled system can be insufficient to obtain a suitable performance. Specifying a lower bound for the decay rate equal 𝛾=2, to obtain a faster transient response and using the LMI (3.7) and (3.8) from Theorem 3.3, and (5.3) from Section 5, one feasible solution was obtained ,.𝑄=90.071−22.22−22.2210.662𝑌=5.4955−3.8158 (6.4) A state-derivative feedback matrix was calculated using (3.9) 𝐾=−0.056149−0.47492. (6.5) For the solution (6.5) one has det(𝐼+𝐵𝐾)=0.056149, and the simulation result of the controlled system for the same initial condition 𝑥(0), is presented in Figure 2. Note that in Figure 2, the settling time was approximately equal to 1 second and max𝑦^′(𝑡)𝑦(𝑡) is equal to 1<𝜉₀=10. Then, the specifications were satisfied by using the designed controller. To facilitate the implementation of the controller, the specification of bounds on the state-derivative feedback matrix 𝐾 can be done using the optimization procedure stated in Lemma 5.1, with 𝜇₀=1. The optimal values, obtained with the “LMI control toolbox” are given in Table 1.
Note that the absolute values of the entries of 𝐾 are smaller than the obtained without optimization method, given in (6.3) and (6.5), respectively.
This procedure can also be applied to the control design of uncertain systems subject to failures.

Table 1 

Figure 1 

The response of the signal $y(t)$ of the controlled system (2.4).

Figure 2 

The response of the signal $y(t)$ of the controlled system (2.4), with bound on the decay rate.

Example 6.2. Consider the linear uncertain descriptor system represented by matrices: 𝐸=00200100−10𝑒₃₃0𝑎0000,𝐴=¹¹0000400−1030010−2, (6.6) where 0.8≤𝑒₃₃≤1.2 and 5.4≤𝑎₁₁≤6.4.
A fail in the actuator is described by:
𝐵=10011𝑏₃₂01, (6.7) where 𝑏₃₂=1 without fail, or 𝑏₃₂=0 with fail of the actuator. Then, the vertices of the polytope are given by triple: (𝐸_𝑖,𝐴_𝑗,𝐵_𝑘)={(𝐸₁,𝐴₁,𝐵₁),(𝐸₁,𝐴₁,𝐵₂), (𝐸₁,𝐴₂,𝐵₁), (𝐸₁,𝐴₂,𝐵₂), (𝐸₂,𝐴₁,𝐵₁), (𝐸₂,𝐴₁,𝐵₂), (𝐸₂,𝐴₂,𝐵₁), (𝐸₂,𝐴₂,𝐵₂)}, where 𝐸₁=00200100−100.800000,𝐸₂=,𝐴00200100−101.200000₁=5.40000400−1030010−2,𝐴₂=,𝐵6.40000400−1030010−2₁=10011001,𝐵₂=.10011101 (6.8)
And the example was solved considering stability with decay rate. It was specified a lower bound for the decay rate equal to 𝛾=2, an output peak bound 𝜉₀=10, and an initial condition 𝑥(0)=[0.30.100]^′. Using LMI control toolbox for solving the set of LMI (4.7) from Theorem 4.2 with (5.3), a feasible solution was the following:
,.𝑄=21.496−1.7143−24.0315.9229−1.71435.2937−1.282−20.904−24.031−1.28275.6345.00445.9229−20.9045.0044268.7𝑌=43.5123.359−135.59−7.66192.3436−7.84812.194218.933 (6.9) A robust state-derivative feedback matrix is obtained using (4.8) 𝐾=0.0680190.34647−1.76720.029854−0.012215−1.7426−1.1708×10⁻⁴−0.064834. (6.10) The locations in the 𝑠-plane of the eigenvalues, for the vertices (𝐸_𝑖,𝐴_𝑗,𝐵_𝑘), of the robust controlled system, are plotted in Figure 3. There exist eight vertices, and four eigenvalues for each vertice. Considering that the output system is 𝐶=10000010, (6.11) the responses of the controlled system with parameter 𝑒₃₃=0.8, and 𝑎₁₁=6.4 for uncertain matrices 𝐸 and 𝐴 respectively, are showed in Figure 4. Note that with (dotted line) or without (solid line) fail of the actuator the controlled system has fast transient responses. Now, solving the optimization procedure stated in Lemma 5.1, with LMI (4.7), (5.3), and 𝜇₀=1, the optimal values, obtained with the “LMI control toolbox” were the following: 𝑄=2.166−0.79427−1.94124.435−0.794271.78390.4143−10.039−1.94120.41437.62810.751714.435−10.0390.751717.0245×10⁶,𝑌=,3.4619−0.22593−12.759−1.43440.98266−2.662−1.570510.009𝛽=178.26,𝐾=0.283560.37604−1.62083.2763×10⁻⁷−0.3028−1.5813−0.19705−6.2275×10⁻⁷. (6.12)(6.13) Note that some absolute values of the entries of 𝐾 in (6.13) are greater than the obtained in first design, given in (6.10). However, the norm of matrix 𝐾 obtained in first design is ‖𝐾‖=1.939 and one obtained from optimization procedure is ‖𝐾‖=1.7655. Therefore the optimization procedure was able to control problem with a smaller norm of the state-derivative feedback matrix 𝐾.

Figure 3 

The eigenvalues location of the vertices from robust controlled uncertain system (4.1) and (2.2), subject to failures.

Figure 4 

The response of the signal $y(t)$ of the controlled system, with and without fail of the actuator.

7. Conclusions

Necessary and sufficient stability conditions based on LMI for state-derivative feedback of linear descriptor systems, were proposed. We can include in the LMI-based control design, the specification of the decay rate, bounds on output peak, and bound on the state-derivative feedback matrix 𝐾. The plant can be linear time-invariant SISO or MIMO, and can also have polytopic uncertainties in its parameters or be subject to structural failures. In this case, one obtains a fault-tolerant design. Therefore, the new design methods allow a broader class of plants and performance specifications, than the related results available in the literature, for instance in [19, 25, 39]. The proposed methods are LMI-based designs that, when feasible, can be efficiently solved by convex programming techniques. Theoretical analysis and numerical simulations illustrate these results.

Acknowledgments

The authors gratefully acknowledge the financial support by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) from Brazil.