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On stabilizability and exact observability of stochastic systems with their applications
Weihai Zhang, Bor-Sen Chen
TL;DR
The paper develops an analogous theory for stochastic systems, addressing stabilizability and observability questions. It introduces new concepts and reports that the PBH criterion still holds for exact observability, while Theorem 7 has applications in system analysis and synthesis.
Problem
Stochastic stabilizability is an interesting problem, motivating development of an analogous theory for stochastic systems.
Method
The paper defines new stochastic-system concepts and uses the spectrum of generalized Lyapunov operators to analyze system properties.
Results
The PBH criterion still holds for exact observability, and Theorem 7 is reported to have important applications in system analysis and synthesis.
Takeaways & Limitations
The results extend deterministic-system ideas to stochastic systems and support applications in system analysis and synthesis.
Takeaways & Limitations
Robust quadratic stabilization remains to be studied for time-varying and nonlinear cases.
Abstract
from arXiv · showhide
This paper discusses the stabilizability, weak stabilizability, exact observability and robust quadratic stabilizability of linear stochastic control systems. By means of the spectrum technique of the generalized Lyapunov operator, a necessary and sufficient condition is given for stabilizability and weak stabilizability of stochastic systems, respectively. Some new concepts called unremovable spectrums, strong solutions, and weakly feedback stabilizing solutions are introduced. An unremovable spectrum theorem is given, which generalizes the corresponding theorem of deterministic systems to stochastic systems. A stochastic Popov-Belevith-Hautus (PBH) criterion for exact observability is obtained. For applications, we give a comparison theorem for generalized algebraic Riccati equations (GAREs), and two results on Lyapunov-type equations are obtained, which improve the previous works. Finally, we also discuss robust quadratic stabilization of uncertain stochastic systems, and a necessary and sufficient condition is given for quadratic stabilization via a linear matrix inequality (LMI).
2 Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan
The paper was completed in 2001, submitted to Automatica, and later published as a brief paper in 2004.
- The paper was completed in 2001 and submitted to Automatica for possible publication.
- It was published as a brief paper in Automatica, volume 40, pages 87–94, in 2004.
- The accompanying material is intended to help readers identify the Automatica publication easily.
1 Introduction
The paper develops spectral characterizations for stabilizability and exact observability in linear stochastic systems, extending deterministic-system ideas while exposing important differences caused by stochastic diffusion control.
- The study focuses on stabilizability and exact observability for linear stochastic controlled systems.The system includes drift and diffusion terms, with feedback potentially entering both through B and D.
- A closed-loop operator spectrum yields necessary and sufficient conditions for stochastic stabilizability and weak stabilizability.The approach defines LK and uses its spectrum to characterize feedback stabilization properties.
- The paper introduces unremovable spectrums, strong solutions, and weakly feedback stabilizing solutions to extend deterministic concepts to stochastic systems.An unremovable spectrum theorem is presented as a stochastic generalization of the deterministic result.
- The stochastic PBH criterion is not sufficient for stabilizability when control enters the diffusion term, although it remains valid for exact observability.This difference is identified as an essential distinction between deterministic and stochastic systems.
- A stochastic PBH criterion is obtained for exact observability, while related observability and detectability implications are clarified.The paper also discusses stochastic detectability by duality.
- Applications include improved results for GAREs and Lyapunov-type equations, plus a necessary and sufficient LMI condition for robust quadratic stabilization of uncertain stochastic systems.
2 Stabilizability and spectrum of stochastic systems
This section characterizes stochastic stabilizability through the spectrum of a generalized Lyapunov operator and identifies unremovable spectra via matrix equalities.
- Mean-square stabilizability is equivalent to finding a feedback gain K such that the closed-loop operator spectrum lies in C−.The operator acts on symmetric matrices and represents the evolution of the state second-moment matrix.
- The closed-loop operator LK maps X to (A + BK)X + X(A + BK)′ + (C + DK)X(C + DK)′.Its induced matrix representation has n(n + 1)/2 variables because X is symmetric.
- An unremovable spectrum is a value λ for which a nonzero symmetric X satisfies the eigenvalue relation for every feedback gain K.
- The unremovable spectrum theorem reduces this feedback-independent condition to three equalities involving A, B, C, and D.These are XA + A′X + C′XC = λX, XB + C′XD = 0, and D′XD = 0.
- When D ≠ 0, the conjectured PBH-style sufficiency for stochastic stabilizability fails, demonstrating a difference from deterministic systems.
3 The spectral characterization for stochastic observability and detectability
The paper applies spectral techniques to characterize stochastic exact observability and detectability, yielding a PBH-type criterion and showing that the two properties are not equivalent.
- Exact observability: Exact observability means every nonzero initial state produces a nonzero output at some positive time.The output is y(t)=Qx(t), where x(t) follows the stochastic differential equation.
- Exact observability: [A, C|Q] is exactly observable if and only if no nonzero symmetric X satisfies XA′ + AX + CXC′ = λX and QX = 0.This is the paper’s stochastic spectral criterion for exact observability.
- Spectral and PBH characterizations: The stochastic criterion reduces exact observability to complete observability of an induced deterministic system and recovers a PBH-based rank characterization.The induced system uses the generalized Lyapunov operator, while the earlier rank test is expressed through P0.
- Detectability: Stochastic detectability is defined by stabilizability of the dual system, and it rules out nonzero solutions of the spectral equation with Re(λ)≥0.The corresponding condition is sufficient in general and becomes necessary under an additional condition on C1.
- Detectability: Exact observability and stochastic detectability have no implication in either direction, although stochastic detectability implies deterministic detectability of (Q,A).The paper gives examples witnessing both non-implication directions and proves the deterministic detectability consequence.
4 On weak stabilizability of stochastic systems
The paper characterizes weak stabilizability through the spectrum of a generalized Lyapunov operator and provides sufficient Lyapunov and LMI conditions.
- Definition and spectral characterization: Weak stabilizability means a state-feedback gain K makes the stochastic closed-loop system weakly stable.The closed-loop dynamics use drift A+BK and diffusion C+DK.
- Definition and spectral characterization: System (1) is weakly stabilizable if and only if the generalized Lyapunov operator has spectrum contained in C−,0 for some feedback gain K.The operator spectrum contains n(n+1)/2 eigenvalues counted in the stated characterization.
- Sufficient conditions: Weak stabilizability follows from either of two Lyapunov-type inequalities or an associated matrix inequality condition.The conditions include inequalities involving P and K, plus an LMI in P and Y.
- Sufficient conditions: The LMI formulation is obtained by setting Y=KP and applying the Schur lemma to the resulting inequality.This connects the feedback variable K with the linearized variables P and Y.
- Consequences and scope: Weak stabilizability of the stochastic system implies weak stabilizability of the deterministic pair (A,B).The paper derives this by taking P=I in the sufficient condition.
- Consequences and scope: The converse of the spectral proposition fails even when σ(A+BK) is contained in C−.The paper explicitly notes this limitation and refers to an example.
5 Some applications
The applications develop comparison and stability results for generalized Riccati and Lyapunov-type equations, clarifying solution strength and stability under observability or detectability assumptions.
- GAREs: The paper identifies a gap in classifying GARE solutions because deterministic closed-loop spectral stability alone does not establish stochastic closed-loop stability.This motivates a feedback-based definition of strong solutions.
- GAREs: A GARE comparison theorem states that under stabilizability and ordered weighting matrices, a maximal solution dominates a given symmetric solution and is strong.The assumptions include R≥R̂, Q≥Q̂, and positivity of the relevant R+D′PD terms.
- GAREs: When Q≥0, R>0, and the system is stabilizable, the GARE has a maximal solution that is also a strong solution.This strengthens the maximal-solution conclusion previously established under the same conditions.
- GAREs: If Q≥0, R≥0, and a GARE solution satisfies P>0, then it is a weakly feedback stabilizing solution.The result follows from the Lyapunov inequality induced by the GARE.
- Lyapunov-type equations: For Lyapunov-type equations, stochastic detectability plus a nonnegative solution implies stability, while exact observability makes a positive solution necessary and sufficient.The stability conclusion is expressed through asymptotic mean-square stability and decay of the state covariance.
6 Robust stabilization of stochastic systems
This section develops robust quadratic stabilization for an uncertain stochastic system and gives a necessary-and-sufficient LMI-based condition, including an explicit stabilizing feedback law.
- Problem formulation: The uncertain stochastic system includes norm-bounded uncertainty ΔA=EFG with F′F≤I and state and control terms in both drift and diffusion.The uncertainty satisfies a matching condition, while the system dynamics are given in equation (70).
- Definition: Quadratic stabilizability requires a positive definite matrix P and α>0 such that the differential generator satisfies L[V(x)]≤−α||x||^2 for all x.The stabilizing control is a state-feedback law u(t)=Kx(t), making the closed-loop system quadratically stable.
- Feedback law: The resulting quadratically stabilizing control law is u(t)=Kx(t)=YX^−1x(t).This feedback gain follows from the change of variables relating K, P, X, and Y.
- Practical implications: Because the condition is an LMI, existing tools can test whether it is feasible, giving Theorem 8 practical value.An analogous theorem is stated for a model with uncertainty in additional system matrices.
7 Conclusion
The conclusion summarizes a spectrum-based theory for stochastic stabilizability and related observability and Riccati-equation results, while identifying open robust-stabilization problems.
- Scope: The paper also addresses exact observability, quadratic stabilization of uncertain stochastic systems, and related GARE and Lyapunov-type problems.The abstract-level conclusion states that further discussion of GAREs, filtering, and stochastic stability is forthcoming.
- Stabilizability: The paper uses spectrum techniques to obtain necessary-and-sufficient conditions for stabilizability and weak stabilizability of stochastic systems.The framework defines the closed-loop operator and reports phenomena differing from deterministic systems.
- New concepts: It introduces unremovable spectrums, strong solutions of GAREs, and weakly feedback stabilizing solutions as concepts in the stochastic theory.These concepts support the paper’s treatment of stabilizability and generalized Riccati equations.
- Applications: The conclusion highlights applications of Theorem 7 in system analysis and synthesis.The supplied passage identifies these applications as important but does not specify their individual forms.
- Open problems: Robust quadratic stabilization remains open for time-varying and nonlinear cases.The conclusion explicitly identifies these cases as topics requiring further study.