Generalized Ratio Control of Discrete-Time Systems

1. Introduction

The problem of the ratio feedback control is one of the specific topics in the theory of control synthesis. It is well practically motivated by applied realizations but not favorable developed in a state control technique or in combination with the state estimation theory. However, a considerable number of problems in the ratio control design have to deal with systems subjected to constraint conditions, which are other than linear, or directly formulated as singular constrained tasks. In the typical case [1, 2] where the system state reflects certain physical entities, constraints usually prescribe the system state, the region of technological conditions. If the ratio control is not formulated as a task with the equality constraints, the application requires further procedures of controlling the evolution of the set-valued ratio. Notably, a special form of the problems can be defined while the system state variables satisfy constraints and interpreted as descriptor systems [3–6]; but, the system with state equality constraints generally does not satisfy the conditions under which the results of descriptor systems can be used. Moreover, if the design task is interpreted as a singular problem, constraint associated methods have to be developed to design the controller.

In principle, it is possible to design the controller that stabilizes a system and simultaneously forces its closed-loop properties to satisfy given constraints [7, 8]. Following the idea of linear quadratic (LQ) control application, these approaches heavily rely on set-valued calculus as well as on min-max theory [9, 10], which are not simple and lead to rather cumbersome technical and numerical procedures. A more simple technique, using equality constraints formulation for discrete-time multiinput/multioutput (MIMO) systems, is introduced in Refs. [11, 12]. Based on the eigenstructure assignment principle, a slight modification of equality constraint technique is presented in Ref. [13].

Many tasks that arise in state-feedback control formulation can be reduced to standard convex problems that involve matrix inequalities. Generally, optimal solutions of such problems can be computed by using the interior point method [14], which converges in polynomial time with respect to the problem size. A review of the progress made in this field can be found in Refs. [15–17] and the references therein. In the given sense, the stability conditions are expressed in terms of linear matrix inequalities (LMI), which have a notable practical interest due to the existence of numerical LMI solvers [18, 19].

The chapter devotes the design conditions to obtain a closed-loop system in which minimally two state variables are rebind by the prescribed ratio. The generalized ratio control principle is reformulated as the full-state feedback control with one equality constraint. Solving this problem, the technique for an enhanced BRL representation [20, 21] is exploited, to circumvent potentially ill-conditioned singular task concerning the discrete-time systems control design with state equality constraints [22]. Due to application of the enhanced BRL, which decouple the Lyapunov matrix and the system matrices, the design task stays well-conditioned. These conditions impose such control that assures asymptotic stability for time-invariant discrete control under defined equality constraints. The presented way, based on projecting the target state variables into a subset of the system state space, adapts the idea of performing the LQ control principle in the fault tolerant control and the constraint control of discrete-time stochastic systems [23, 24].

The outline of this chapter is as follows. Continuing the introduction outlines in Section 1, the problem formulation is principally presented in Section 2. Section 3 is dedicated to the mathematical backgrounds supporting the problem solution and the exploited discrete-time LMI modifications are given in Section 4. These results are used in Section 5 to examine the linearization problems in bilinear matrix inequalities, so that in Section 5, these results can be given with convex formulation of control design condition, guaranteeing a feasible solution of the generally singular design task. Subsequently, numerical examples to illustrate basic properties of the proposed method are presented in Section 6, and Section 7 is finally devoted to a brief concluding remarks.

Throughout the chapter, the following notations are used: x^T and X^T denote the transpose of the vector x and matrix X, respectively, for a square matrix X < 0 that X is a symmetric negative definite matrix, the symbol I_n represents the nth order unit matrix, Y ^⊝1 denotes the Moore-Penrose pseudoinverse of a nonsquare Y, ∥ ⋅ ∥ represents the Euclidean norm for vectors and the spectral norm for matrices, IRdenotes the set of real numbers and IR^n × r the set of all n × r real matrices.

2. Problem formulation

Through this chapter, the task is concerned with design of the full-state feedback control to discrete-time linear dynamic systems in such a way that the closed-loop system state variables are constrained in the prescribed ratio. The systems are defined by the set of state equations

where q(i) ∈ IRⁿ is the vector of the state variables, u(i) ∈ IR^r is the vector of the input variables, y(i) ∈ IR^m is the vector of the output variables, and nominal system matrices F ∈ IR^n × n, G ∈ IR^n × r, and C ∈ IR^m × n are real matrices, and i ∈ Z₊.

The discrete transfer function matrix of dimension m × r, associated with the system Eqs. (1) and (2) is defined as

where I_n ∈ IR^n × n is the identity matrix, ỹ(z) and ũ(z) stand for the Z transform of m dimensional output vector and r dimensional input vector, respectively, and a complex number z is the transform variable of the Z transform [25].

In practice, the ratio control maintains the relationship between two state variables [26, 27] and is defined for all i ∈ Z as

Assuming the parameter vector e_h, the task can be expressed by using the system state vector q(i + 1) as

where

It is evident that the generalized ratio control can be defined by a composed structure of e, as well as by a structured matrix E [28].

The task formulated above means the design problem that can be generally defined as the stable closed-loop system synthesis using the linear full-state feedback controller of the form

where K ∈ IR^r × n is the controller feedback gain matrix, and the design constraint is considered in the general matrix equality form

with E ∈ IR^p × n, rank E = p ≤ r. In general, the matrix E reflects prescribed fixed ratio of two or more state variables. The equality Eq. (9) evidently implies ΛEq(i + 1) = 0, where Λ ∈ IR^p × p is an arbitrary matrix.

It is considered in the following the discrete-time system is controllable and observable that is, rankzI−F,G=n∀z∈Cand rankzI−FT,CT=n∀z∈C, respectively [29], and that all state variables are measurable.

3. Basic preliminaries

Proposition 1. (Matrix pseudoinverse) Let Θ is a matrix variable and A, B, and Π are known nonsquare matrices of appropriate dimensions such that

Then all solution to Θ means that

where

while A^⊝ 1 is the left Moore-Penrose pseudoinverse of A, B^⊝ 1 is the right Moore-Penrose pseudoinverse of B and Θ° is an arbitrary matrix of appropriate dimension.

Proof. (see, e.g., Ref. [15])

Proposition 2. Let Ξ ∈ IR^n × n is a real square matrix with nonrepeated eigenvalues, satisfying the equality constraint

then one from its eigenvalues is zero, and the (normalized) vector e^T is the left raw eigenvector of Ξ associated with the zero eigenvalue.

Proof. If Ξ ∈ IR^n × n is a real square matrix satisfying the above given eigenvalues limitation, then the eigenvalue decomposition of Ξ takes the following form

where n_l is the right eigenvector and mlTis the left eigenvector associated with the eigenvalue z_l of Ξ, and {z_l, l = 1, 2, … n} is the set of the eigenvalues of Ξ. Then Eq. (13) can be rewritten as follows:

If eT=mhT, then orthogonal property Eq. (15) implies

and it is evident that Eq. (17) can be satisfied only if z_h = 0. This concludes the proof.□

Proposition 3. (Quadratic performance) Given a stable system of the structure Eqs. (1) and (2), then it yields

where γ_∞ ∈ IR is the H_∞ norm of the transfer function matrix of the system Eq. (3).

Proof. Since Eq. (3) implies

then, evidently,

where ∥ H(z) ∥ is H₂ norm of the discrete transfer function matrix H(z).

Since the H_∞ norm property states

using the notation ∥ H(z) ∥_∞ = γ_∞, then Eq. (21) can be naturally rewritten as

Thus, based on the Parseval’s theorem, Eq. (22) gives

and using squares of the elements, the inequality Eq. (23) subsequently results in

Thus, Eq. (24) implies Eq. (18). This concludes the proof.□

If it is not in contradiction with other design constraints, Eq. (18) can be used as the extension to a Lyapunov function candidate for linear discrete-time systems, since it is positive.

4. Quadratic performances

The above presented assumptions are imposed to obtain LMI structures exploiting H_∞ norm, known as the bounded real lemma LMIs. To simplify proofs of theorems in following, proof sketches of the BRL are presented, since more versions of BRL can be constructed.

Proposition 4. (Bounded real lemma) The autonomous system Eqs. (1) and (2) is stable with the quadratic performance γ_∞, if there exist a symmetric positive definite matrix P ∈ IR^n × n and a positive scalar γ_∞ ∈ IR such that

where I_r ∈ IR^r × r and I_m ∈ IR^m × m are identity matrices, respectively.

Hereafter, ∗ denotes the symmetric item in a symmetric matrix.

Proof. (compare, e.g., Refs. [16] and [23]) Defining the Lyapunov function candidate as follows:

then Eq. (18) implies that with the H_∞ norm γ_∞ of the transform function matrix Eq. (3), the inequality Eq. (27) is positive. The forward difference of Eq. (27) along a solution of the autonomous system Eq. (1) can be written as

and, using the description of the state system Eqs. (1) and (2), the inequality Eq. (28) becomes

Thus, introducing the notation

it is obtained

where

Since, using the Schur complement property with respect to the matrix element γ∞−1CTC, Eq. (32) can be rewritten as

then, applying the dual Schur complement property, Eq. (33) implies Eq. (26). This concludes the proof.□

Direct application of the second Lyapunov method [30, 31] and BRL in the structure given by Eqs. (25) and (26) for affine uncertain systems as well as in constrained control design is in general ill-conditioned owing to singular design conditions [13]. To circumvent this problem, an enhanced LMI representation of BRL is proposed, where design condition proof is based on another form of LMIs.

Proposition 5. (Enhanced LMI representation of BRL) The autonomous system Eqs. (1) and (2) is stable with the quadratic performance γ_∞, if there exist a symmetric positive definite matrix P ∈ IR ^n × n, a regular square matrix Q ∈ IR ^n × n, and a positive scalar γ_∞ ∈ IR such that

where I_r ∈ IR ^r × r and I_m ∈ IR ^m × m are identity matrices.

Proof. Since, Eq. (1) can be rewritten as

with an arbitrary square matrix Q ∈ IR ^n × n, it yields

Now, not substituting Eq. (1) into Eq. (28), but adding Eq. (37) and its transposition to Eq. (28), it can be obtained that

Thus, considering Eq. (2), then Eq. (38) can be rewritten as

where

and

Since Eq. (41) can be written as

then, using the dual Schur complement property, Eq. (43) can be transformed in the form

To obtain a LMI structure visually comparable with Eq. (26), the following block permutation matrix is defined

Then, premultiplying the left side of Eq. (43) by T_a° and postmultiplying the right side of Eq. (43) by the transposition of T_a° lead to the inequality in Eq. (35). This concludes the proof.□

It is evident that Lyapunov matrix P is separated from the matrix parameters of the system F, G, and C, i.e., there are no terms containing the product of P and any of them. By introducing the slack variable matrix Q, the product forms are relaxed to new products QF and QG, where Q needs not be symmetric and positive definite. This enables a robust BRL, which can be obtained to deal with linear systems with parametric uncertainties, as well as with singular system matrices.

Considering a symmetric positive definite matrix Q ∈ IR^n × n, the following symmetric enhanced LMI representation of BRL is evidently obtained.

Corollary 1. (Enhanced symmetric LMI representation of BRL) The autonomous system Eqs. (1) and (2) is stable with the quadratic performance γ_∞, if there exist symmetric positive definite matrices P, Q ∈ IR ^n × n and a positive scalar γ_∞ ∈ IR such that

where I_r ∈ IR ^r × r, I_m ∈ IR ^m × m are identity matrices.

Note, Corollary 1 provides the identical condition of existence to Proposition 4, if the equality P = Q is set.

5. Control law parameter design

The state-feedback control problem is finding, for an optimized (or prescribed) scalar γ > 0, the state-feedback gain K such that the control law guarantees an upper bound of γ_∞ of the closed-loop transfer function, while the closed-loop is stable. Note, all the above presented BRL structures applied in the control law synthesis lead to bilinear matrix inequalities and have to be linearized.

Theorem 1. System Eqs. (1) and (2) under control Eq. (3) is stable with quadratic performance γ_∞, if there exist a positive definite symmetric matrix R ∈ IR ^n × n, a matrix Y ∈ IR ^r × n, and a positive scalar γ_∞ ∈ IR such that

When these inequalities are satisfied, the control law gain matrix is given as

Proof. Since P is positive definite, the transform matrix T_∞ can be defined as follows:

Then, premultiplying the left side of Eq. (35) and postmultiplying the right side of Eq. (35) by T_∞ gives

Inserting F ← F_c = (F − GK) into Eq. (51) gives

and with

Eq. (53) implies Eq. (48). This concludes the proof.□

Theorem 2. System Eqs. (1) and (2) under control Eq. (3) is stable with quadratic performance γ_∞, if there exist positive definite symmetric matrices S, O ∈ IR ^n × n, a matrix Y ∈ IR ^r × n, and a positive scalar γ_∞ ∈ IR such that

When these inequalities are satisfied, the control law gain matrix is given as

Proof. Considering that Q is positive definite, the transform matrix T∞∘can be defined as follows:

Therefore, premultiplying the left side of Eq. (46) and postmultiplying the right side of Eq. (46) by the matrix T∞∘gives

Substituting F ← F_c = (F − GK) into Eq. (58) gives

and with

Eq. (59) implies Eq. (55). This concludes the proof.□

6. Ratio control design

Using the control law Eq. (3), the closed-loop system equations take the form

Prescribed by a matrix E ∈ IR^p × n, rank E = p ≤ r, it is considered the design constraint Eq. (9) for all nonzero natural numbers i. From Proposition 2, it is clear that such kind of design is a singular task, where Eq. (9) gives

which evidently implies

Evidently, the equality

can be satisfied, as well as the closed-loop system matrix F_c = F − GK has to stable (all its eigenvalues are from the unit circle in the complex plane Z).

Lemma 1.The equivalent state-space description of the system Eqs. (1) and (2) under control Eq. (3), in which closed-loop state variables satisfying the condition Eq. (9) is

where

while L ∈ IR^r × r is the projection matrix (the orthogonal projector of EG onto the null space NEG[23]) and K° ∈ IR^r × n is the ratio control gain matrix.

Proof. Premultiplying the left side of Eq. (65) by the identity matrix, it yields

which implies the particular solution

where

is the left Moore-Penrose pseudoinverse of EG.

Using the equality Eq. (65), then Eq. (69) can be also written as

which implies

respectively, where I_r ∈ IR^p × p is the identity matrix. It is evident that Eq. (74) can be satisfied only if

Thus, Eq. (11) implies all solutions of K as follows

where K° is an arbitrary matrix with appropriate dimension, and evidently Eq. (76) gives Eq. (68). This concludes the proof.□

Considering the model involving the given ratio constraint on the closed-loop system state variables Eqs. (66)–(68), the design conditions are presented in the following theorems.

Theorem 3. System Eqs. (1) and (2) under the control (3), and satisfying the constraint Eq. (4) is stable with the quadratic performance γ_∞, if there exist positive definite matrices S, O ∈ IR ^n × n, a matrix Y° ∈ IR ^r × n, and a positive scalar γ_∞ ∈ IR such that

When these inequalities are satisfied, the control law gain matrices are given as

where J, L are defined in Eq. (68).

Proof. Substituting Eq. (68) into Eq. (59) gives

Using the notation

Eq. (80) implies Eq. (78). This concludes the proof.□

The ratio control does not exclude a forced regime given by the control law

where w(i) ∈ IR^m is desired output signal vector and W ∈ IR^m × m is the signal gain matrix. Using the static decoupling principle, the conditions to design the signal gain matrix W can be proven.

Lemma 2. If the system Eqs. (1) and (2) is square, which is stabilizable by the control policy Eq. (82) and Ref. [32]

then the matrix W takes the form

where I_n ∈ IR^n × n is the identity matrix.

Proof. In a steady state, the system equations Eqs. (1) and (2), and the control law Eq. (82) imply

where q_o, w_o are the steady-state values of the vectors q(i), w(i), respectively. Since from Eq. (85), it can be derived that

and

considering y_o = w_o, Eq. (87) implies Eq. (84). This concludes the proof.□

Theorem 4. If the closed-loop system state variables satisfy the state constraint Eq. (63), then the common state variable vector q_d(i) = Eq(i), q_d(i) ∈ IR^k attains the steady-state value

Proof. Using the control policy Eq. (82), then

Since K satisfies Eq. (65), then Eq. (89) implies

and it is evident that the tied state variable q_d(i) of the closed-loop system in a steady state is proportional to the steady state of the desired signal w_o and takes the value Eq. (88). This concludes the proof.□

7. Illustrative examples

To demonstrate properties of proposed approach, the classical example for a helicopter control [33] is taken, where the discrete-time state-space representation Eqs. (1) and (2) for the sampling period Δt = 0.05s consists of the following parameters

The state constraint, defining the ratio control of two state system variables, is specified as

and subsequently it yields

Solving Eqs. (77) and (78) using self-dual-minimization (SeDuMi) package for Matlab [19], the feedback gain matrix design problem in the constrained control is feasible with the results

Inserting Y° and S into Eq. (79), the gain matrix K° is computed as

and Eq. (79) implies the full-state feedback gain matrix values

It can be easily verified that the closed-loop system matrix takes the format

while the ratio control law rises up the stable closed-loop system with the closed-loop system matrix eigenvalues spectrum

Note that one from the resulting eigenvalue of F_c is zero (rank(E) = 1)), because Proposition 2 prescribes this constrained design task as a singular problem. Using the connection between the eigenvector matrix N and M as given by Eq. (17), it is possible to show that this instance is documented also by the structure of M, while

where the structure of the third row of M correspondents to the structure of the constraint vector E, while a4=m3T1/m3T4=−1.5.

To illustrate the closed-loop system property in the forced mode, the signal gain matrix W is computed by using Eq. (84) as follows

Therefore, according to Theorem 4, the constraint given on the states of the system under study is satisfied with zero offset in the autonomous regime and with offset value equal q_dw in the forced mode, i.e.,

while

The simulation results of the closed-loop system response in the autonomous and forced mode are presented, where Figure 1 is concerned with the system state variables response in the autonomous regime and Figure 2 with the system state variables response in the forced mode. It is evident that the condition Eq. (9) is satisfied at all time instant, except initial time instant in the above given way (see the time response of the additive of variable, which is included as q_d(i) in the figures).

For comparison, an example is given for default design of state feedback gain matrix using BRL structure of LMIs. Solving Eqs. (54) and (55), the task is feasible with the Lyapunov matrix variables

and parameter matrix variable

Therefore, using Eq. (56), the nominal control law gain matrix K is computed as

the closed-loop system matrix takes the form

while the closed-loop system matrix eigenvalues spectrum is

To apply in the forced mode, the signal gain matrix W is now computed by using Eq. (84) as follows:

The simulation results of the nominal closed-loop system response are illustrated in Figures 3 and 4, where Figure 3 is concerned with the system state variables response in the autonomous regime and Figure 4 with the system state variables response in the forced mode.

Since these two control structures are of interest in the context of full-state control design, matching the presented results, it is evident that the system dynamics in both cases are comparable.

8. Concluding Remarks

In this chapter, an extended method is presented, based on the classical memoryless feedback H_∞ control principle of discrete-time systems, if the ratio control is reformulated by an equality constraint setting on associated state variables. The asymptotic stability of the control scheme is guaranteed in the sense of the enhanced representation of BRL, while resulting LMIs are linear with respect to the system state variables, and does not involve products of the Lyapunov matrix and the system matrix parameters, which provides one way of solving the singular LMI problem. Moreover, formulated as a stabilization problem with the full-state feedback controller, the control gain matrix takes no special structure. The formulation allows to find a solution without restrictive assumptions and additional specifications on the design parameters. It is clear from Theorem 4 that the control law strictly solves the problem even in the unforced mode. The validity of the proposed method is demonstrated by numerical examples.

Acknowledgements

The work presented in this chapter was supported by VEGA, the Grant Agency of the Ministry of Education and the Academy of Science of Slovak Republic, under Grant No. 1/0608/17. These supports are very gratefully acknowledged.