New robust stable MPC using linear matrix inequalities

New robust stable MPC using linear matrix inequalities

M.A. Rodrigues and D. Odloak

Chemical Engineering Department, University of São Paulo (USP),

P. O. Box 61548, 05424-970, São Paulo - SP, Brazil,

E - mail: [e-mail]

(Received: June 2, 1999; Accepted: August 31, 1999)

Abstract - This paper addresses the stability of Model Predictive Control (MPC) with output feedback. The proposed controller uses a new state-space formulation of the system, and the control problem is presented as an LMI optimization problem. The stability condition for the closed loop is included as a Lyapunov inequality. The resulting optimization problem becomes nonlinear with the inclusion of the stabilizing condition. A suboptimal solution is developed and the problem reduces to a pair of coupled LMI problems. An iterative solution that converges to a stable output feedback gain is proposed. A polytopic set of process models can be considered. A simulation example is included in the paper and shows that the proposed strategy eliminates the usual practice of enforcing robustness by detuning the MP controller.

Keywords: model predictive control, output feedback, linear matrix inequalities, robust stability.

INTRODUCTION

Chemical processes are frequently multivariable with strong interactions between the process variables. These processes are moderately linear and usually exhibit slow dynamics and time delays. In addition, their operating conditions usually require the presence of constraints on some of the process variables. Since chemical processes are often large scale, the possibility of including economic objectives that can provide optimal operation of the plant is very attractive. All of these characteristics are easily handled by Model Predictive Control (MPC) strategy. In fact, MPC is the advanced control framework most frequently used in the chemical processes industries. Moreover, in the last decade there has been a large number of papers focusing on MPC in the process control literature.

In general, the MPC strategy selects an open-loop sequence of control moves that minimizes an objective function, subject to constraints on the input (manipulated variables) and outputs (controlled variables) or states. This objective function is usually defined as the square of the difference between the predicted system outputs and the required setpoints. A term weighting the input moves is often added to this objective function. The system dynamics can be linear or nonlinear. This optimization problem is reevaluated at each sampling time with new plant measurements. More details on the basic concepts of MPC can be seen in Garcia et al. (1989) and Bequette (1991).

A classic result in process modeling is that a single linear model usually provides a deficient representation of the real process under operating conditions different from those under which the model was identified. Consequently, MPC that considers only a nominal linear model may present a poor performance when applied to a real system (Morari and Zafiriou, 1989). It is important to design a controller that takes into account model uncertainties due to unmodeled dynamics, different operating conditions and disturbances.

Regarding the MPC literature we can verify that stability and robustness are relevant characteristics, considering the number and diversity of applications that have appeared lately. However, the number of papers related to the nominal stability of MPC is expressively larger than the number of papers concerning robustness. Focusing on nominal MPC stability, Muske and Rawlings (1993) and Rawlings and Muske (1993) have shown that an infinite horizon MPC globally stabilizes the system. MPCs with an infinite horizon have also been studied by Zheng et al. (1995) and Zheng and Morari (1995). In a different approach, Keerthi and Giltbert (1988) investigated the use of constraints on the terminal states in order to stabilize nominal nonlinear Receding Horizon Control (RHC). A unified approach to infinite horizon MPC and MPC with end constraints was proposed by Zheng (1997). In this paper, Zheng showed that input and output weighting matrices can guarantee the stability of the closed loop if they are made variable along the prediction and control horizons. Recently, Chen and Allgöwer (1998a,b) presented new results for the stability of nonlinear MPC using terminal state constraints.

As previously cited, the robust MPC literature is less extensive than the literature related to nominal stability. Zafiriou (1990) and Zafiriou and Marchal (1991) made use of the contraction principle to provide the robust stability of QDMC with input and output constraints. Polak and Yang (1993) proposed a Moving Horizon Control (MHC) that guaranteed robust stability by a contraction inequality constraint. Badgwell (1997a) generalized the nominally stable controller proposed by Rawlings and Muske (1993) to the case of an uncertain model that belongs to a set of possible models. An alternative to treat the robustness of MPC is given by the use of "min-max" algorithms. In these methods the worst case cost function for a set of possible models is minimized. Following this approach, Scokaert and Mayne (1998) proposed a min-max MPC where the stability condition is obtained by the inclusion of end constraints.

In the field of optimal control and H_µ control, a new and promising technique for the robust synthesis of stable controllers, which is based on Linear Matrix Inequalities (LMI) has been studied. A tutorial on these developments can be seen in Boyd et al. (1994). One of the main advantages of the LMI tools is their computational skills, since the LMI solvers make use of powerful interior point numerical algorithms. Moreover, the representation of inequality constraints becomes trivial with the LMI methods. The use of LMI in MPC was initially proposed by Kothare et al. (1996), which presented a min-max constrained infinity horizon MPC written as an LMI optimization problem. In the Kothare et al. paper, model uncertainty was considered by polytopic model representation and by parametric uncertainties. They developed an upper bound for the cost function considering the set of all plant models to provide stability to the system. However, their work considered that the states of the system are known at any sampling instant, which makes the approach very restrictive for practical purposes. Most of the process control applications deal with systems where only a few of the states are measured to constitute the system output that fed the MP controller. Then, in the real case we usually have the controller working in an output feedback structure instead of the state feedback as assumed by Kothare et al. (1996).

The solution of control problems using LMI can be considered a field open to contributions, and this paper represents a new step in the development of synthesis of the robust stable MPC. Here the output feedback problem is analyzed. The model of the system is given by a convenient state-space representation and the control problem is written as an LMI optimization problem. The concepts used here stem from the Lyapunov theory for the quadratic stabilization of discrete-time dynamic systems. A guaranteed closed-loop stability condition is derived and it corresponds to a Lyapunov-type constraint, which results in a nonlinear matrix inequality (NLMI). Thus, to overcome this difficulty a numerical iterative algorithm is proposed and the NLMI is converted into a pair of coupled LMIs. A polytopic set is considered for the uncertain model. The algorithm has guaranteed convergence, as long as it is given an initial controller gain that stabilizes the nominal model and all polytope vertices.

The paper is organized as follows. In the next section, the state-space model representation is presented and the unconstrained MPC problem is formulated as an LMI problem. Then the guaranteed stability condition is derived as a nonlinear matrix inequality constraint. An iterative numerical algorithm is proposed and the nonlinear optimization problem is decomposed into two coupled LMI problems. Afterwards, it is shown that the MP controller obtained is robustly stable and the convergence properties of the proposed algorithm are discussed. Then follows description of the extension of the formulation to incorporate input constraints. In the last section, a system from the process control literature is used to illustrate the proposed controller and to compare it to the classical DMC algorithm.

SYSTEM REPRESENTATION AND MP CONTROL LAW

The majority of MPC controllers applied in the processes industries use a step or impulse response model to predict the system output. However, a state-space model of the process is necessary for control purposes, since the existing analytical machinery for the analysis and synthesis of the closed loop is usually based on this model representation. Several different state-space models for MPC have been proposed in the literature by Li et al. (1989), Morari and Lee (1991) and Tvrzská de Gouvêa and Odloak (1997). In this section, we present a new state-space model that is suitable for the synthesis of the MP control law. This model is based on the well-known ARX model, and the resulting structure of the MP controller is presented next.

The State-Space Model

The process model considered here is based on the following open-loop input-output model representation for a discrete linear multivariable (MIMO) system with nu inputs and ny outputs:

where

is the model output at sampling instant k.

is the control move at sampling instant k.

D = 1- z ^-1, D Î Â is the difference operator.

The state variables are chosen as follows:

where

is the state corresponding to the output of the system at prediction instant k+ i.

is the matrix of the step response coefficients calculated at time i = 1, 2, k, ... ,n.

n is the prediction horizon.

Moreover, n is such that

with A_k = 0 for k > nA.

Matrix S_i can be evaluated from (1) by the following relation:

S_j = S(j)

where S(j) satisfies

A(z-1) S(j) = B(z-1) I( j)

and

Thus, it is straightforward to show that

Equations (2) and (3) can be written in vector form as

where

The prediction of the system output, given by (6), can be corrected considering the difference between the plant measurement, given by , and the model output at the same time. The correction term is given by

where

is the real plant output available at sampling instant k+1.

is the plant state at sampling instant k+1;

is the prediction horizon for the real plant.

R is the state observer gain. In the usual MPC approach, R has the following form:

.

Thus, the corrected state vector can be written as

Substituting (5) and (9) into (10), the following equation can be obtained:

It is reasonable to assume that the true model of the plant has the same structure as the model used in the controller. Thus, we assume the true model of the plant is given by:

Substituting (12) into (11), the following equation is obtained:

It is more convenient to represent the state as deviation variables that can be obtained by subtracting the vector of output setpoints from both sides of equation (13) as follows:

The same procedure can be applied to the real plant, except that in this case it is not necessary to correct the state with plant measurements. The resulting equation is

We can include equations (14) and (15) in a single equation as follows

which is in the standard form of the usual state-space model and can be written as follows:

is the vector of deviations of the augmented state (model + plant);

Thus, the deviation of the controlled variables can be obtained by the following expression:

Suppose now that the MP control strategy results in the following output feedback law:

where K_mpcis the controller gain whose calculation procedure will be described in sequel.

Substituting equation (19) and (20) into equation (17), results in

where is the closed loop system state matrix.

Description of Model Uncertainties

After establishing the model structure, it is important to describe the system uncertainties. It can be noted that matrices F and Y of equation (17) completely define the process model. Since the real plant model is not precisely known, then some elements of these matrices are also unknown. There are several ways of characterizing model uncertainties (see, e.g., Badgwell, 1997b). Here, the polytopic representation has been adopted. This approach considers that the model that truly represents the system belongs to a convex set W , defined by (Colaneri et al., 1997):

co {•}denotes convex hull built from matrices , i = 1, 2, ..., L, where L is the number of vertices of the polytope. This means that any matrix may be represented by a linear combination of the extreme matrices that constitute the polytope vertices, , i = 1, 2, ..., L :

.

The models, i = 1, 2, ..., L, can be obtained by (Kothare et al., 1996):

(a) the identification of a linear model at different operating conditions ;

(b) the linearization of the nonlinear model, that describes the system dynamic at different operating points. This is possible as long as the linearization conditions, given by the Hartman-Gröbman theorem (Guckenheimer and Holmes, 1983), are verified.

The MPC in the LMI Form

In the previous sections, it the structure of the state-space model and the framework to represent model uncertainty were established. Now we focus on the controller gain, K_mpc, and we derive the optimization problem whose solution produces this gain. It is important to note that in equation (17), Du(k)corresponds to the control move at the first instant on the control horizon. However, in the MPC approach the controller gain results from the minimization of a cost function over a control horizon with m instants. Therefore, the equation for the state prediction has to be modified in order to take into account the control moves at instants k, k + 1, ..., k + m - 1. The resulting prediction equation is

or, in the deviation form

Now suppose that the input is obtained by the following output feedback law:

Substituting equation (27) into equation (24), results in equation (28):

In order to derive the control law, now consider the following MPC cost function

where X and G are positive definite weighting matrices.

Using equations (27) and (28), it is straightforward to show that

Thus, the optimization problem that defines the unconstrained MPC is the following:

min g

g, Km

subject to

J (k) - g < 0

or

such that g > 0, g Î Â.

Applying to (31) Schur’s complement presented as Lemma 2 of the Appendix, and choosing the following block matrices:

then the MPC optimization problem is converted in the following LMI optimization problem:

Problem P1)

min g

g, Km

s. t.: (32)

Problem P1) can be solved by an available LMI solver, such as MATLAB LMI Toolbox (Gahinet et al., 1995) and LMISol (Oliveira et al., 1997).

PROVIDING ROBUST STABILITY TO MPC VIA LMI

In the last section, the MPC problem was written as an unconstrained LMI optimization problem, whose solution gives the controller gain. The resulting control input is based solely on a single process model that is usually called the nominal model. Also, there is no guarantee that the controller will stabilize the nominal system in closed loop. In this section we derive a stability condition for the proposed controller that stabilizes not only the nominal system, but all the systems whose models are contained in W.

Deriving a Stabilizing Constraint for Problem P1)

Consider the closed-loop state-space model given by equation (22) applied to each of the extreme points of the polytopic set defined by equation (23) as follows

Applying the Lyapunov stability theorem for discrete-time linear systems (Åström and Wittenmark, 1990) to the system defined by (33), the following inequality is obtained

:,i = 1,...,L

Substituting equation (33) into (34) results in

which is equivalent to

are given by (18) and (19) but here they are evaluated for each vertex of the polytope.

Inequality (35) represents the Lyapunov stability condition that will be included in the MPC problem presented in the preceding section. The theorem that follows states a necessary and sufficient condition for the stability of the uncertain MPC controller. These arguments stem from the theory of quadratic stability of discrete-time systems (Geromel et al., 1991).

Theorem 1 – Stabilizing Condition for the Uncertain System

The controller gain obtained by the solution of the output feedback MPC via LMI, stated as problem P1), will stabilize the system described by (17) to (19) if and only if it is possible to find a Lyapunov matrix, P, that satisfies the inequality (35) for any i Î {1, 2, ..., L }.

Proof:

It is assumed that the unknown model of the system is a convex combination of known models,

i.e.,

.

Thus, Lyapunov stability condition for this system is

Carrying out the matrix products on the left-hand side (LHS) of (36), the following expression is obtained:

This expression can be written in the following form:

Substituting into (37) results in

Applying the property of convex sets of matrices established in Lemma 1 presented in the Appendix, the following expression is obtained:

It is easy to show that inequality (38) is equivalent to

Therefore, it is clear that the system will be stabilized by the MP controller if inequality (35) holds and the proof of the theorem is concluded.

This theorem establishes an important result for uncertain systems, since it is shown that if the Lyapunov stability condition is verified for each vertex of W , then any plant whose model belongs to polytope W will be stabilized by K_mpc, which is obtained from the solution of problem P1).

Stabilizing MPC via LMI

If the complement of Schur is applied to inequality (35), the following inequality is obtained:

,

Thus, if for each vertex i of the polytope W inequality (40) is included in the MPC problem P1), the resulting controller will stabilize all the models contained in W. In the robustly stable output feedback MPC problem, the unknown variables are the Lyapunov matrix, P, and the controller gain, K_mpc. However, inequality (40) is nonlinear because it involves the product of P and K_mpc. For this reason, the inclusion of the stabilizing constraint in the MPC, written as an LMI optimization problem, makes the solution of this problem a difficult task.

In the case of state feedback, this problem is solved by the use of convenient parameterizations (Kothare et al., 1996; Geromel et al., 1998; Syrmos et al., 1997). Such an approach can be better understood with the help of the following arguments:

In inequality (35), substitute P = Q^-1, such that Q>0:

Multiplying inequality (41) on the left by -Q^T and on the right by Q,, inequality (42) is obtained:

Applying the complement of Schur to inequality (42) results in

Now, if we define

Y = K_mpc C Q

it is possible to eliminate the nonlinearity of (43). However, the question is how to recover a unique K_mpc? The inversion is not possible since matrix C is not a square matrix in the case of output feedback. Several numerical algorithms have been proposed to overcome similar difficulties (Geromel et al., 1998; Iwasaki et al., 1994; Geromel et al., 1996; Rodrigues and Odloak, 1999), particularly in the fields of optimal and H_µ control. In the following section, we propose a new iterative algorithm that allows the inclusion of the stability constraint in MPC. This algorithm provides a suboptimal solution for the resulting MPC optimization problem.

Algorithm for the Robust Stable MPC

This algorithm allows the inclusion of the Lyapunov stability constraint represented by inequality (40) in the optimization problem labeled as problem P1). The algorithm can be described by the following steps:

1.0) Initialize the counter (k = 0) and the auxiliary variable g_k = 0;

2.0) Find an initial gain, K_mpc0, such that all closed-loop eigenvalues of each polytope vertex are strictly inside the unit circle;

3.0) Solve the following optimization problem, labeled P2), in order to obtain a symmetric positive definite matrix P.

Problem P2)

max a

a, P

such that.:

,i = 1, 2, ..., L (44)

a > 0

P > 0

with a Î Â;

3.1) Let P_k = Pand go to step 4.0;

4.0) With the Lyapunov matrix, P, calculated in step 3.0, find g that minimizes the problem labeled as P3).

Problem P3)

min a

g, Km

such that:

5.0) i) If the absolute value of the difference between the previous and present values of the objective function of problem P2) is less than the convergence parameter, , then apply the control move corresponding to the first line of K_m evaluated at this iteration and go to the next sampling instant. Here

ii) If this condition does not hold, go to step 6.0);

6.0) Update the counter, k = k + 1, and the auxiliary variable, g_k = g . Let K_mpc= N_sK_m, and return to step 3.0).

In the algorithm, it has been assumed that it is always possible to obtain a gain that initially stabilizes all the model and all plants constituting uncertainty domain W. Perhaps this assumption may seem too restrictive. However, for open-loop stable systems, the null gain is always an initial feasible solution of problem P2). Consequently, for this case it is always possible to solve the nonlinear constraint converted into a pair of coupled LMI. From the solution of P2), a Lyapunov matrix that maximizes the margin of stability for the given controller gain is obtained. In the sequence of the algorithm, a new K_mpc is obtained by the solution of P3), subject to the stability constraint (45) evaluated with matrix P calculated in P2). This procedure is repeated until the convergence criterion is reached. The resulting control move is implemented and the algorithm goes to the next sampling time.

Theorem 2 - Convergence Properties

The iterative algorithm described in the previous section provides a sequence of feasible solutions such that g_k+1£ g_k , k = 0, 1, .

Proof: At each iteration of the algorithm, problem P2) is solved for a feasible K_mpck that obeys inequality (45) for a given P_k. Corresponding to P_k one has g_k, and K_mpck , which results from the solution of P3). The new solution of P2) gives P_k+1. Since with P = P_k+1, K_mpck is a feasible solution to P3), the new solution of P3) gives g_k+1£ g_k which concludes the proof of the theorem.

INPUT CONSTRAINTS

Input constraints are usually considered in the application of MPC to chemical processes. These constraints are easily included in the controller in the LMI approach. The constraint considered here is the maximum bound on the move of manipulated variables:

Du^T N_IDu £ Du²_{j, max} (46)

where

and is the quadratic maximum bound on the move of the j^th manipulated variable.

Applying the complement of Schur to inequality (46), the following LMI is obtained:

Therefore, inequality (47) should be included in problem P3) if we need to prevent the control move from being larger then a maximum allowable bound.

NUMERICAL EXAMPLE

This section presents the results of simulations that illustrate the application of the proposed Robust Stable MPC via LMI (RSMPC). In the numerical examples the process considered is based on a classic system from the MPC literature, the Shell benchmark problem (Prett and Garcia, 1988) reduced to a case with two inputs and two outputs. The time delays of the system were modified to values that are more realistic. For the implementation of this example, we used the Matlab LMI Control Toolbox(Gahinet et al., 1995).

The system considered here is given by the following transfer function:

The controlled variables (outputs) are the top product endpoint (y₁) and the bottom reflux temperature (y₂) of the FCC main fractionator. The manipulated variables (inputs) are the flow of the side draw (u₁) and the bottom reflux heat duty (u₂).

The objective of the first simulation is to compare the RSMPC with the unconstrained DMC. In this simulation it is assumed that the model used by the controller and the plant are known but that they are different. The model used by the controller is the nominal model given by equation (48). The model of the plant is obtained by changing the gains and time constants of the nominal model. The resulting model of the plant is the following:

The tuning parameters used by RSMPC and DMC are the following:

and the convergence parameter of RSMPC is e = 1x10^-3.

RSMPC is initialized with a gain equal to half of the DMC gain, i.e., K_mpc0 = o.5 K_DMC. For this simulation and the others in the sequel, the servo control problem for a unit step change in the setpoint is considered.

Figure 1 shows that the two controllers can stabilize the system and the time responses of the inputs and outputs of the system are almost the same for RSMPC and DMC. This shows that the suboptimal solution obtained with the RSMPC procedure reproduces very closely the DMC solution at any sampling time.

Figure 2 presents the norm of the feedback gains of RSMPC and DMC for the simulation shown in Figure 1. Since in this example no input constraints are considered, the DMC solution is optimal for the objective function defined in equation (30). However, almost the same responses in terms of system inputs and outputs can be obtained for controllers with different gains. When the closed loop is stable with DMC, the procedure followed by RSMPC tends to produce gains with larger norms than DMC.

In the other simulation carried out with this system, the behavior of RSMPC is studied with modeling uncertainties given by a three-vertex polytope. The responses of RSMPC are also compared to the responses of DMC. The model used by RSMPC and the adopted plant model are obtained by convex combinations of the polytope vertices and consequently both are inside the polytope. The model of the controller and the true plant model are described in Table 1. DMC uses the same model, plant and tuning parameters considered by RSMPC. Table 2 presents the models that characterize the polytope that defines convex set W. These are obtained by changing the gains and time constants of the model represented in equation (48).

The tuning parameters of the controller are the following:

Figure 3 shows the responses of the system with RSMPC for a unit step change in the output setpoints and the settings described above. It can be observed that the robust controller performs satisfactorily and stabilizes the system without any difficulty. It should also be noted that the input weights (G ) are set equal to zero, showing that RSMPC does not need this tuning parameter, which in the conventional MPC is usually applied to stabilize the system. Figure 4 shows that, under the same conditions, DMC is unstable.

The norm of the feedback gain of RSMPC and DMC is also represented (Figure 5). Since DMC is unstable, it is obvious that its gain can not be optimal. We can also accept that for this case, controllers with larger gains than DMC tend to destabilize the system in closed loop. This can explain the results obtained with RSMPC, which show a gain smaller than the DMC gain for most of the sampling instants.

CONCLUSION

This paper presents a new solution to the problem of robust stability of MPC, with model uncertainty represented by a polytopic set. The method uses a state-space model of the system in the MP controller. The predictive control problem is presented as an LMI optimization problem that takes into account uncertainties in the model of the system. A necessary and sufficient condition for robust stability of the MP controller applied to the uncertain system is transformed into a nonlinear matrix inequality. An iterative numerical algorithm that converts this nonlinear matrix inequality into a pair of linear matrix inequalities is proposed. Provided there is a feasible initial feedback gain, the iterative algorithm produces a sequence of decreasing values of the cost function. The procedure makes use of conventional and powerful LMI solvers. A typical process is used to illustrate the application of the proposed MP controller. The results show that RSMPC is able to keep the process under control even when DMC fails. The example also shows that the usual practice of enhancing robustness by detuning the controller via an increase in input weights may be minimized. In this case, the work of tuning the controller would be simplified because the number of parameters is reduced, since the input weights can be removed from the set of controller parameters. Moreover, the results presented here enforce the guaranteed improvement in the stability of the proposed controller, showing that when DMC exhibits an unstable behavior, RSMPC stabilizes the system with a feedback gain whose norm is smaller than the DMC norm.

ACKNOWLEDGMENTS

Support for this research was provided by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under grant 96/08087-0 and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under grant 300860/97-8.

NOMENCLATURE

A, B matrices of the process model, eq. (1)

F

, Y matrices of the state-space model for MPC, eq. (17)

F

_cl state matrix of the closed loop, eq. (22)

C matrix that defines the output of the state-space model for MPC, eq. (17)

e augmented state deviation from setpoint

G process transfer function

K_mpc MPC feedback gain

L number of vertices of the polytope

M state matrix of the prediction model

m control horizon

state matrix of the true plant model

N_s auxiliary matrix used to collect the first nu control moves

n prediction horizon

n_A order of matrix A

n_B order of matrix B

nu number of inputs

ny number of outputs

P Lyapunov stability matrix

R disturbance correction gain, eq. (9)

S step response coefficient matrix of the predictive model

true plant step response coefficient matrix

dynamic matrix of MPC, eq (24)

T sampling time

u manipulated input

x state of the system, eq. (5)

state of the true plant, eq. (12)

z system output

Greek symbols

a

objective function of problem

P2)

e

RSMPC convergence parameter

g

objective function of problem

P3)

G

input weighting matrix, eq. (29)

X

output weighting matrix, eq. (29)

q

time delay

Abbreviations

ARX Auto Regressive Exogenous Model

DMC Dynamic Matrix Control

LMI Linear Matrix Inequality

MHC Moving Horizon Controller

MIMO Multi-Input Multi-Output

MP Model Predictive

MPC Model Predictive Control

NLMI Nonlinear Matrix Inequalities

QDMC Quadratic Dynamic Matrix Control

RHC Receding Horizon Control

RSMPC Robust Stable Model Predictive Control

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APPENDIX

The following lemma presents an important matrix property that is used in the proof of

Theorem 1.

Lemma 1 - Convex Matrices Property

With these assumptions the following matrix inequality holds:

With these assumptions the following matrix inequality holds:

Proof:

1. Multiplying the matrices on the left-hand side (

   LHS) of inequality (A.1), adding and subtracting terms

results in

Since the bracket terms of LHS are nonnegative and the rest of this expression is equal to the right-hand side of (A.1), it can be concluded that LHS is always less than or equal to zero. Then the proof is concluded.

(b) The proof for this case can be obtained by induction of that presented in (a).

Lemma 2 – The Complement of Schur

Given the block symmetric matrix F ,

where submatrices F ₁₁ and F ₂₂ are symmetric. Thus, the following statements hold:

i) If F ₁₁ is positive definite,

ii) If F ₂₂ is positive definite,

Proof:

Consider the following positive definite matrix:

Thus, the identity above can be shown:

which proves part i) of the lemma.

By following the same procedure, but using matrix T defined above

we can show part ii) of this lemma, which completes the proof.

Publication Dates

• Publication in this collection
  24 Apr 2000
• Date of issue
  Mar 2000

History

• Received
  02 June 1999
• Accepted
  31 Aug 1999