Advanced LMI Techniques in Control System Design

In our previous article, we introduced the fundamentals of Linear Matrix Inequalities (LMIs) and their application in controller design.

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This follow-up explores some advanced LMI techniques including Schur's lemma, variable elimination methods, and practical implementation with MATLAB code examples.

Schur's Lemma: A Key Tool for LMI Problems

Schur's lemma (also known as Schur complement) is an essential technique for reformulating nonlinear matrix inequalities into equivalent LMI forms. Consider the following nonlinear matrix inequality:

\begin{equation} \left[\begin{array}{cc} Q(x) & S(x) \\ S(x)^T & R(x)\end{array}\right] \gt 0 \end{equation}

Where Q(x) , S(x) , and R(x) are affine matrix functions of the variable . If $R(x) \gt 0$ , then according to Schur's lemma, this inequality is equivalent to:

\begin{equation} R(x) \gt 0, \quad Q(x) - S(x)R(x)^{-1}S(x)^T \gt 0 \end{equation}

However, the second condition involves $R(x)^{-1}$ , which is not an affine function. Schur's lemma allows us to convert this back to an LMI form:

\begin{equation} \left[\begin{array}{cc} Q(x) & S(x) \\ S(x)^T & R(x)\end{array} \right] \gt 0 \Leftrightarrow \begin{cases} R(x) \gt 0 \\ Q(x) - S(x)R(x)^{-1}S(x)^T \gt 0 \end{cases} \end{equation}

This equivalence is extremely useful for reformulating optimization problems into LMI form. For example, in our previous article, we encountered the term Z^T Z which could be linearized using Schur's lemma.

Variable Elimination in LMI Problems

Often in control design, we encounter matrix variables that need to be eliminated to obtain an LMI in terms of the primary variables of interest. The Elimination Lemma (also known as Projection Lemma) is particularly useful for this purpose.

Consider the LMI problem:

\begin{equation} \Psi + M \Theta N^T + N \Theta^T M^T \lt 0 \end{equation}

where $\Psi, M, N$ are known matrices and $\Theta$ is the variable matrix we want to eliminate. The Elimination Lemma states that there exists a $\Theta$ such that the inequality holds if and only if:

\begin{equation} W_M^T \Psi W_M \lt 0 \text{ and } W_N^T \Psi W_N \lt 0 \end{equation}

where W_M and W_N are matrices whose columns form bases for the null spaces of M^T and N^T , respectively.

This lemma is particularly useful in controller synthesis problems where we often need to eliminate controller variables to get a solvable LMI formulation.　

Practical Implementation with MATLAB

Let's implement some of these concepts using MATLAB's Robust Control Toolbox. First, we'll solve the $H_\infty$ control problem from our previous article:

% System matrices
A = [0 1 0; 0 0 1; 2 -1.2 2.3];
B = [0; 0; 1];
B2 = [5; 1; -2];
C = [-1 5 0];

% Define LMI variables
setlmis([]);
P = lmivar(1, [3 1]);  % Symmetric positive definite matrix
gamma = lmivar(1, [1 1]);  % Scalar gamma
Z = lmivar(2, [1 3]);  % Controller matrix K = Z*P^-1

% LMI #1: P \gt 0
lmiterm([-1 1 1 P], 1, 1);  % P \gt 0

% LMI #2: The main H_infinity condition (using Schur complement)
lmiterm([2 1 1 P], A, 1, 's');  % PA + A'P
lmiterm([2 1 1 Z], -B, 1, 's');  % -BZ - Z'B'
lmiterm([2 1 2 P], 1, B2);  % PB2
lmiterm([2 1 3 0], C');  % C'
lmiterm([2 2 2 0], -1);  % -I
lmiterm([2 3 3 gamma], -1, 1);  % -gamma*I

% Create and solve the LMI system
lmisys = getlmis;
[copt, xopt] = mincx(lmisys, [0 1 0*ones(1,size(Z,2)*size(Z,1))]);

% Extract the solution
Pval = dec2mat(lmisys, xopt, P);
Zval = dec2mat(lmisys, xopt, Z);
gamma_opt = dec2mat(lmisys, xopt, gamma);

% Compute the controller
K = Zval/Pval;

disp(['Optimal gamma: ', num2str(sqrt(gamma_opt))]);
disp('Controller K:');
disp(K);

% Verify closed-loop stability
Acl = A - B*K;
eig(Acl)

S-Variable Approach for Multi-Objective Control

For more complex control problems involving multiple objectives, the S-variable approach provides an elegant framework. This approach introduces additional variables to convert bilinear matrix inequalities (BMIs) into LMIs.

Consider a state-feedback control problem where we want to simultaneously achieve:

$H_\infty$ performance bound
Pole placement in a specific region
Input constraint satisfaction

Generalized KYP Lemma for Frequency-Domain Specifications

The Generalized Kalman-Yakubovich-Popov (KYP) lemma allows us to specify control requirements in specific frequency ranges instead of across all frequencies. This is particularly useful for targeted controller design.

For a frequency interval $[\omega_1, \omega_2]$ , we can express frequency-domain constraints using the LMIs:

References on Generalized KYP Lemma

Iwasaki, T., & Hara, S. (2005). Generalized KYP lemma: unified frequency domain inequalities with design applications. IEEE Transactions on Automatic Control, 50(1), 41-59. - The seminal paper that introduced the generalized KYP lemma, establishing the connection between finite-frequency domain inequalities and LMIs.
Rantzer, A. (1996). On the Kalman-Yakubovich-Popov lemma. Systems & Control Letters, 28(1), 7-10. - A foundational paper providing a simplified proof of the KYP lemma that paved the way for its generalizations.
Iwasaki, T., Meinsma, G., & Fu, M. (2000). Generalized S-procedure and finite frequency KYP lemma. Mathematical Problems in Engineering, 6(2-3), 305-320. - An important paper that extends the KYP lemma to handle finite frequency ranges using the S-procedure.

Robust Control with Polytopic Uncertainty

When system parameters are uncertain but bounded within a polytope, we can design a robust controller using LMIs. Consider a system with polytopic uncertainty:

\begin{equation} \dot{x} = A(\lambda)x + B(\lambda)u \end{equation}

where $A(\lambda) = \sum_{i=1}^L \lambda_i A_i$ and $B(\lambda) = \sum_{i=1}^L \lambda_i B_i$ with $\sum_{i=1}^L \lambda_i = 1$ , $\lambda_i \geq 0$ .

Common matrix is used to design controller for the system with polytopic uncertainty.

References

Skelton, R. E., Iwasaki, T., & Grigoriadis, K. (1998). A Unified Algebraic Approach to Linear Control Design. Taylor & Francis. - A comprehensive resource that covers structured controller design using LMIs, including decentralized control.
Langbort, C., Chandra, R. S., & D'Andrea, R. (2004). Distributed control design for systems interconnected over an arbitrary graph. IEEE Transactions on Automatic Control, 49(9), 1502-1519. - Presents techniques for designing structurally constrained controllers for networked systems.
Zecevic, A. I., & Siljak, D. D. (2010). Control of complex systems: Structural constraints and uncertainty. Springer. - A comprehensive book on controlling complex systems with structural constraints, including decentralized approaches using LMIs.
Petersen, I. R. (2006). A linear matrix inequality approach to robust control of a class of uncertain interconnected systems. International Journal of Robust and Nonlinear Control, 16(4), 161-177. - Addresses robust decentralized control for interconnected systems with uncertainties.
Ghaoui, L. E., Oustry, F., & AitRami, M. (1997). A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control, 42(8), 1171-1176. - Introduces techniques for solving non-convex problems arising in structured controller design.

Dynamic Output Feedback Using LMIs

For cases where state feedback is not possible, we can design a dynamic output feedback controller using LMIs. This involves finding matrices , , , , and that satisfy certain LMIs:

Key References on Dynamic Output Feedback Control Using LMIs

Scherer, C., Gahinet, P., & Chilali, M. (1997). Multiobjective output-feedback control via LMI optimization. IEEE Transactions on Automatic Control, 42(7), 896-911. - The seminal paper that introduced the LMI framework for multi-objective synthesis of linear output-feedback controllers, establishing the foundation for modern dynamic output feedback design.
Gahinet, P. (1996). Explicit controller formulas for LMI-based H∞ synthesis. Automatica, 32(7), 1007-1014. - Presents explicit controller formulas for output feedback design that avoid numerical issues in controller reconstruction.
Oliveira, M. C., Geromel, J. C., & Bernussou, J. (2002). Extended H₂ and H∞ norm characterizations and controller parametrizations for discrete-time systems. International Journal of Control, 75(9), 666-679. - Introduces extended LMI characterizations that improve numerical conditioning of the dynamic output feedback design problem.
de Oliveira, M. C., Bernussou, J., & Geromel, J. C. (1999). A new discrete-time robust stability condition. Systems & Control Letters, 37(4), 261-265. - Presents important dilated LMI conditions that are fundamental to modern dynamic output feedback controller design.
Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics. - The foundational textbook that presents the basic theory of LMIs and their applications to control problems, including output feedback design.
Apkarian, P., & Gahinet, P. (1995). A convex characterization of gain-scheduled H∞ controllers. IEEE Transactions on Automatic Control, 40(5), 853-864. - Extends the LMI approach to parameter-varying systems with important insights for dynamic output feedback design.
Chilali, M., & Gahinet, P. (1996). H∞ design with pole placement constraints: an LMI approach. IEEE Transactions on Automatic Control, 41(3), 358-367. - Shows how to incorporate regional pole placement constraints in dynamic output feedback design.
Geromel, J. C., Bernussou, J., & de Oliveira, M. C. (1999). H₂-norm optimization with constrained dynamic output feedback controllers: decentralized and reliable control. IEEE Transactions on Automatic Control, 44(7), 1449-1454. - Addresses decentralized output feedback control design using LMIs.
Kočvara, M., & Stingl, M. (2003). PENNON: A code for convex nonlinear and semidefinite programming. Optimization Methods and Software, 18(3), 317-333. - Describes numerical methods for solving the non-convex optimization problems that arise in output-feedback controller design.

Recent Developments and Alternative Approaches

Kanev, S., Scherer, C., Verhaegen, M., & De Schutter, B. (2004). Robust output-feedback controller design via local BMI optimization. Automatica, 40(7), 1115-1127. - Presents a local BMI optimization approach for robust output-feedback controller design, addressing the inherent non-convexity of the problem.
Veenman, J., & Scherer, C. W. (2014). A synthesis framework for robust gain-scheduling controllers. Automatica, 50(11), 2799-2812. - Presents a framework applicable to structured dynamic output feedback synthesis using dilated LMI conditions.
Rosinová, D., & Veselý, V. (2008). Output feedback controller design: Non-iterative LMI approach. Journal of Electrical Engineering, 59(6), 341-343. - Proposes a non-iterative LMI procedure for output feedback controller design with guaranteed cost.
Pipeleers, G., Demeulenaere, B., Swevers, J., & Vandenberghe, L. (2009). Extended LMI characterizations for stability and performance of linear systems. Systems & Control Letters, 58(7), 510-518. - Introduces extended LMI characterizations that reduce conservatism in output feedback design.
Zečević, A. I., & Šiljak, D. D. (2010). Control of complex systems: Structural constraints and uncertainty. Springer. - Comprehensive book on controlling complex systems with structural constraints, including dynamic output feedback approaches using LMIs.
Apkarian, P., Noll, D., & Alazard, D. (2008). Controller design via nonsmooth multidirectional search. SIAM Journal on Control and Optimization, 47(2), 728-743. - Presents a non-convex optimization approach for fixed-order output feedback controller design.

Conclusion

In this article, we've explored several advanced techniques for using LMIs in control system design, including Schur's lemma for transforming nonlinear constraints into LMIs, variable elimination to simplify complex problems, and practical implementations for various control objectives.

The mathematical elegance of LMIs combined with efficient numerical solvers makes them an extremely powerful tool for modern control engineering. By formulating control problems as LMIs, we can systematically address multiple design objectives and constraints simultaneously, leading to controllers with proven performance guarantees.

For further exploration, we recommend experimenting with the MATLAB code examples provided and adapting them to your specific control problems. The versatility of LMIs means they can be applied to a wide range of control challenges across different domains.

References

Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). "Linear Matrix Inequalities in System and Control Theory." SIAM Studies in Applied Mathematics.
Scherer, C., and Weiland, S. (2000). "Linear Matrix Inequalities in Control." Lecture Notes, Dutch Institute for Systems and Control.
Chilali, M., and Gahinet, P. (1996). "H∞ design with pole placement constraints: an LMI approach."IEEE Transactions on Automatic Control, 41(3), 358-367.
Iwasaki, T., and Hara, S. (2005). "Generalized KYP lemma: unified frequency domain inequalities with design applications."IEEE Transactions on Automatic Control, 50(1), 41-59.

Software Tools and Resources

Several software tools are available for solving multi-objective LMI problems:

YALMIP - A MATLAB toolbox for modeling and solving optimization problems, with excellent support for LMI problems.
SeDuMi - A popular optimization package for solving problems with linear, quadratic and semidefiniteness constraints.
MATLAB Robust Control Toolbox - Provides tools for designing controllers for systems with uncertain parameters using LMI-based approaches.
Rcsdp - An R interface to the CSDP semidefinite programming library for users who prefer R over MATLAB.
CVXOPT - A Python package for convex optimization that can handle LMI problems.

Advanced LMI Techniques in Control System Design

Schur's Lemma: A Key Tool for LMI Problems

Further Reading and References

Variable Elimination in LMI Problems

Further Reading and References

Practical Implementation with MATLAB

S-Variable Approach for Multi-Objective Control

Further Reading and References

Generalized KYP Lemma for Frequency-Domain Specifications

References on Generalized KYP Lemma

Robust Control with Polytopic Uncertainty

References

Dynamic Output Feedback Using LMIs

Key References on Dynamic Output Feedback Control Using LMIs

Recent Developments and Alternative Approaches

Conclusion

References

Software Tools and Resources

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