Abstract
For linear uncertain systems, we consider a finite-horizon \(H_{\infty }\) control problem. A weight matrix of the control cost in the functional of this problem is singular. In this case, the Riccati equation approach is not applicable to solution of the considered \(H_{\infty }\) problem, meaning that it is singular. To solve this problem, a regularization method is proposed. Namely, the original problem is associated with a new \(H_{\infty }\) control problem for the same dynamics, while with a new functional. The weight matrix of the control cost in the new functional is a nonsingular parameter-dependent matrix, which becomes the original weight matrix for zero value of the parameter. For all sufficiently small values of this parameter, the new \(H_{\infty }\) control problem is regular, and it is a partial cheap control problem. Using its asymptotic analysis, a controller solving the original singular \(H_{\infty }\) control problem is designed. An illustrative example is presented.
Keywords
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Doyle, J., Francis, B., Tannenbaum, A.: Feedback Control Theory. Macmillan Publishing Co., London (1990)
Basar, T., Bernard, P.: \(H_{\infty }\)-Optimal Control and Related Minimax Design Problems: A Dynamic Games Approach. Birkhauser, Boston (1991)
Petersen, I., Ugrinovski, V., Savkin, A.V.: Robust Control Design Using \(H_{\infty }\) Methods. Springer, London (2000)
Martynyuk, A.A., Martynyuk-Chernienko, Yu.A.: Uncertain Dynamical Systems: Stability and Motion Control. CRC Press, Florida (2011)
Glizer, V.Y., Turetsky, V.: Robust Controllability of Linear Systems. Nova Science Publishers, New-York (2012)
Chang, X.H.: Robust Output Feedback \(H_{\infty }\) Control and Filtering for Uncertain Linear Systems. Springer, Berlin (2014)
Fridman, L., Poznyak, A., Rodrguez, F.J.B.: Robust Output LQ Optimal Control via Integral Sliding Modes. Birkhuser, Basel (2014)
Petersen, I.R., Tempo, R.: Robust control of uncertain systems: classical results and recent developments. Autom. J. IFAC 50, 1315–1335 (2014). https://doi.org/10.1016/j.automatica.2014.02.042
Yedavalli, R.K.: Robust Control of Uncertain Dynamic Systems: A Linear State Space Approach. Springer, New York (2014)
Doyle, J.C., Glover, K., Khargonekar, P.P., Francis, B.: State-space solution to standard \(H_{2}\) and \(H_{\infty }\) control problem. IEEE Trans. Autom. Control 34, 831–847 (1989). https://doi.org/10.1109/9.29425
Petersen, I.R.: Disturbance attenuation and \(H_{\infty }\) optimization: a design method based on the algebraic Riccati equation. IEEE Trans. Autom. Control 32, 427–429 (1987). https://doi.org/10.1109/TAC.1987.1104609
Stoorvogel, A.A.: The singular minimum entropy \(H_{\infty }\) control problem. Syst. Control Lett. 16, 411–422 (1991). https://doi.org/10.1016/0167-6911(91)90113-S
Copeland, B.R., Safonov, M.G.: A zero compensation approach to singular \(H^2\) and \(H^{\infty }\) problems. Int. J. Robust Nonlinear Control 5, 71–106 (1995). https://doi.org/10.1002/rnc.4590050202
Xin, X., Guo, L., Feng, C.: Reduced-order controllers for continuous and discrete-time singular \(H^{\infty }\) control problems based on LMI. Autom. J. IFAC 32, 1581–1585 (1996). https://doi.org/10.1016/S0005-1098(96)00103-3
Gahinet, P., Laub, A.J.: Numerically reliable computation of optimal performance in singular \(H_{\infty }\) control. SIAM J. Control Optim. 35, 1690–1710 (1997). https://doi.org/10.1137/S0363012994269958
Orlov, Y.V.: Regularization of singular \(H_2\) and \(H_\infty \) control problems. In: Proceedings of the 36th IEEE Conference on Decision and Control, vol. 4, pp. 4140–4144. IEEE Press, New York (1997). https://doi.org/10.1109/CDC.1997.652517
Stoorvogel, A.A., Trentelman, H.L.: The quadratic matrix inequality in singular \(H_\infty \) control with state feedback. SIAM J. Control Optim. 28, 1190–1208 (1990). https://doi.org/10.1137/0328064
Stoorvogel, A.A.: The \(H_\infty \) Control Problem: A State Space Approach (e-book). University of Michigan, Ann-Arbor (2000)
Chuang, N., Petersen, I.R., Pota, H.R.: Robust \(H^\infty \) control in fast atomic force microscopy. In: Proceedings of the 2011 American Control Conference, pp. 2258–2265. IEEE Press, New York (2011). https://doi.org/10.1109/ACC.2011.5991155
Glizer, V. Y., Kelis, O.: Solution of a singular control problem: a regularization approach. In: Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics, Vol. 1, pp. 25–36. Scitepress - Science and Technology Publications, Setbal, Portugal (2017). https://doi.org/10.5220/0006397600250036
Stoorvogel, A.A., Trentelman, H.L.: The finite-horizon singular \(H_{\infty }\) control problem with dynamic measurement feedback. Linear Algebra Appl. 187, 113–161 (1993)
Glizer, V.Y.: Finite horizon \(H_{\infty }\) cheap control problem for a class of linear systems with state delays. In: Proceedings of the 6th International Conference on Integrated Modeling and Analysis in Applied Control and Automation, pp. 1–10. CAL-TEK SRL Press, Rende, Italy (2012)
Bell, D.J., Jacobson, D.H.: Singular Optimal Control Problems. Academic Press, New-York (1975)
Kurina, G.A.: On a degenerate optimal control problem and singular perturbations. Soviet Math. Doklady 18, 1452–1456 (1977)
Glizer, V.Y.: Solution of a singular optimal control problem with state delays: a cheap control approach. In: Reich, S., Zaslavski, A.J. (eds.) Optimization Theory and Related Topics, Contemporary Mathematics Series, vol. 568, pp. 77–107. American Mathematical Society, Providence (2012). https://doi.org/10.1090/conm/568/11278
Glizer, V.Y.: Stochastic singular optimal control problem with state delays: regularization, singular perturbation, and minimizing sequence. SIAM J. Control Optim. 50, 2862–2888 (2012). https://doi.org/10.1137/110852784
Glizer, V.Y.: Singular solution of an infinite horizon linear-quadratic optimal control problem with state delays. In: Wolansky, G., Zaslavski, A.J. (eds.) Variational and Optimal Control Problems on Unbounded Domains, Contemporary Mathematics Series, vol. 619, pp. 59–98. American Mathematical Society, Providence (2014). https://doi.org/10.1090/conm/619/12385
Turetsky, V., Glizer, V.Y., Shinar, J.: Robust trajectory tracking: differential game/cheap control approach. Int. J. Syst. Sci. 45, 2260–2274 (2014). https://doi.org/10.1080/00207721.2013.768305
Shinar, J., Glizer, V.Y., Turetsky, V.: Solution of a singular zero-sum linear-quadratic differential game by regularization. Int. Game Theory Rev. 16, 1440007-1–1440007-32 (2014). https://doi.org/10.1142/S0219198914400076
Glizer, V.Y., Kelis, O.: Solution of a zero-sum linear quadratic differential game with singular control cost of minimiser. J. Control Decis. 2, 155–184 (2015). https://doi.org/10.1080/23307706.2015.1057545
Glizer, V.Y.: Nash equilibrium in a singular two-person linear-quadratic differential game: a regularization approach. In: Proceedings of 24th Mediterranean Conference on Control and Automation, pp. 1041–1046. IEEE Press, New York (2016). https://doi.org/10.1109/MED.2016.7535851
Glizer, V.Y., Kelis, O.: Singular infinite horizon zero-sum linear-quadratic differential game: saddle-point equilibrium sequence. Numer. Algebra Control Optim. 7, 1–20 (2017). https://doi.org/10.3934/naco.2017001
Glizer, V.Y.: Solution of a singular \(H_{\infty }\) control problem for linear systems with state delays. In: Proceedings of the 2013 European Control Conference, pp. 2843–2848. IEEE Press, New York (2013)
Glizer, V.Y.: Asymptotic analysis and solution of a finite-horizon \(H_{\infty }\) control problem for singularly-perturbed linear systems with small state delay. J. Optim. Theory Appl. 117, 295–325 (2003). https://doi.org/10.1023/A:1023631706975
Glizer, V.Y., Fridman, L., Turetsky, V.: Cheap suboptimal control of an integral sliding mode for uncertain systems with state delays. IEEE Trans. Automat. Control 52, 1892–1898 (2007). https://doi.org/10.1109/TAC.2007.906201
Vasil’eva, A.B., Butuzov, V.F., Kalachev, L.V.: The Boundary Function Method for Singular Perturbed Problems. SIAM Books, PA (1995)
Glizer, V.Y.: Correctness of a constrained control Mayer’s problem for a class of singularly perturbed functional-differential systems. Control Cybernet. 37, 329–351 (2008)
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Appendices
Appendix A: Proof of Theorem 2
The proof consists of four stages.
Stage 1. At this stage, we transform the HIPCCP with \(u(t)=u_{\varepsilon ,0}^{*}[z(t),t]\) to an equivalent \(H_{\infty }\) problem. Remember that the controller \(u_{\varepsilon ,0}^{*}[z(t),t]\), given by (42), solves the HIPCCP if the inequality
is fulfilled along trajectories of the system (7) for all \(w(t)\in L^{2}[0,t_f; E^{m}]\).
Substituting the controller \(u_{\varepsilon ,0}^{*}[z(t),t]\) into the system (7) and the functional (14), as well as using of the block representations for the matrices B(t), D(t), \(G(t)+{\mathcal E}\), A(t) and for the vector z(t) (see equations (10), (12), (15), (23) and (40)), we obtain the following system and functional:
where
Due to (49), the inequality (47) is equivalent to the inequality
along trajectories of the system (48) for all \(w(t)\in L^{2}[0,t_f; E^{m}]\). The latter means that the solvability of the HIPCCP by \(u(t)=u_{\varepsilon ,0}^{*}[z(t),t]\) is equivalent to the solvability of the \(H_{\infty }\) problem (48)–(49), (53).
Stage 2. At this stage, we derive solvability conditions of the \(H_{\infty }\) problem (48)–(49), (53). Consider the terminal-value problem for the Riccati matrix differential equation with respect to the matrix \(\widehat{P}(t)\) in the interval \([0,t_f]\):
We are going to show that the existence of the solution \(\widehat{P}(t,\varepsilon )\) to the problem (54) for a given \(\varepsilon \in (0,\varepsilon _{0}]\) in the entire interval \([0,t_f]\) yields the fulfilment of the inequality (53).
For a given \(\varepsilon \in (0,\varepsilon _{0}]\), consider the Lyapunov-like function
Based on (55), consider the function \(V\big [z_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big ),t,\varepsilon \big ]\). Since \(z_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big )\) is the solution of the initial-value problem (7) with \(u(t)=u_{\varepsilon ,0}^{*}[z(t),t]\), then it is the solution of the initial-value problem (48). Differentiating the function \(V\big [z_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big ),t,\varepsilon \big ]\) with respect to t, and using (48), (54) and (55) yield
Using the function \(w^{*}\big (t,\varepsilon ;w(\cdot )\big ) {\mathop {=}\limits ^{\triangle }} \gamma ^{-2}F^{T}(t)\widehat{P}(t,\varepsilon )z_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big ),\) and taking into account the expression for \(S_{w}(t)\) (see Eq. (17)), we can rewrite (56) as:
From the Eq. (57), we directly obtain the inequality for all \(t\in [0,t_f]\):
Integration of this inequality from \(t=0\) to \(t=t_f\) and use of (55) yield
meaning the fulfilment of the inequality (53) along trajectories of the system (48) for all \(w(t)\in L^{2}[0,t_f; E^{m}]\). This completes the proof of the statement that the existence of the solution \(\widehat{P}(t,\varepsilon )\) to the problem (54) in the entire interval \([0,t_f]\) guarantees the fulfilment of the inequality (53) along trajectories of (48) for all \(w(t)\in L^{2}[0,t_f; E^{m}]\).
Stage 3. At this stage, we show the existence of the solution \(\widehat{P}(t,\varepsilon )\) to the problem (54) in the entire interval \([0,t_f]\). Similarly to the problem (16), (18), we look for the solution of the terminal-value problem (54) in the block form
where the matrices \(\widehat{P}_{1}(t,\varepsilon )\), \(\widehat{P}_{2}(t,\varepsilon )\) and \(\widehat{P}_{3}(t,\varepsilon )\) have the dimensions \((n-r+q)\times (n-r+q)\), \((n-r+q)\times \left( r-q\right) \) and \(\left( r-q\right) \times \left( r-q\right) \), respectively. Substitution of the block representations for the matrices \(S_{w}(t)\), \(\widehat{A}(t,\varepsilon )\), \(\widehat{D}(t)\) and \(\widehat{P}(t,\varepsilon )\) (see Eqs. (17), (50), (58)) into the problem (54) converts this problem into the following equivalent problem:
Looking for the zero-order asymptotic solution of the problem (59)–(61) in the form \(\widehat{P}_{i,0}(t,\varepsilon ) = \widehat{P}_{i,0}^{o}(t) + \widehat{P}_{i,0}^{b}(\tau )\), \((i = 1,2,3)\), \(\tau = (t - t_f)/\varepsilon \), we obtain similarly to Sect. 5.2
Furthermore, the terms of the outer solution \(\widehat{P}_{i,0}^{o}(t)\), \((i=1,2,3)\), satisfy the following terminal-value problem for \(t\in [0,t_{f}]\):
Using that \(\widehat{A}_{4}(t,0) = -P_{3,0}^{o}(t) = - \big (D_{2}(t)\big )^{1/2}\), \(\widehat{D}_{3}(t) = 2D_{2}(t)\), we obtain the unique symmetric positive definite solution of (65)
Using (66) and the fact that \(\widehat{A}_{2}(t) = A_{2}(t)\), \(\widehat{A}_{3}(t,0)= -\big (P_{2,0}^{o}(t)\big )^{T}\), \(\widehat{D}_{2}(t) = P_{1,0}^{o}(t)A_{2}(t)\), one directly has from (64)
Substitution of (67) into (63) and use of (51)–(52) yields the terminal-value problem for \(\hat{P}_{1,0}^{o}(t)\)
It is verified directly by substitution of \(P_{1,0}^{o}(t)\) into (68) instead of \(\widehat{P}_{1,0}^{o}(t)\) and using (33), (34), (36) that the terminal-value problem (68) has a solution on the entire interval \([0,t_f]\) and this solution equals to \(P_{1,0}^{o}(t)\), i.e.,
Moreover, due to the linear and quadratic dependence of the right-hand side of the differential equation in (68) on \(\widehat{P}_{1,0}^{o}(t)\), this solution is unique.
For the terms \(\widehat{P}_{2,0}^{b}(\tau )\) and \(\widehat{P}_{3,0}^{b}(\tau )\), similarly to Sect. 5.2, we have the problem
Using (33), (51), (66), (67), we obtain the unique solution of this problem
This solution satises the inequalities
where \(c>0\) and \(\beta >0\) are some constants.
Now, based on the Eqs. (62), (66), (67), (69), (70) and the inequalities (71), we obtain (similarly to Lemma 2) the existence of a positive number \(\varepsilon _{1}^{*}\) such that, for all \(\varepsilon \in (0,\varepsilon _{1}^{*}]\), the problem (59)–(61) has the unique solution \(\big \{\widehat{P}_{1}(t,\varepsilon ),\widehat{P}_{2}(t,\varepsilon ),\widehat{P}_{3}(t,\varepsilon )\big \}\) in the entire interval \([0,t_f]\). Since the problem (54) is equivalent to (59)–(61), then it has the unique solution (58) in the entire interval \([0,t_f]\) for all \(\varepsilon \in (0,\varepsilon _{1}^{*}]\). The latter, along with the results of the Stages 1 and 2 of this proof, means that the controller \(u_{\varepsilon ,0}^{*}[z(t),t]\), given by (42), solves the HIPCCP, i.e., the inequality (47) is fulfilled.
Stage 4. Using the equations (8), (14), (15), (42), (43), we obtain
This equation, along with the inequality (47), directly yields the inequality (44). This completes the proof of Theorem 2.
Appendix B: Proof of Theorem 3
The proof is based on an auxiliary lemma.
Auxiliary Lemma
Let, for a given \(\varepsilon >0\), \(n\times n\)-matrix-valued function \(\varPhi (t,s,\varepsilon )\), \(0\le s \le t \le t_f\), be the fundamental solution of the system \(dz(t)/dt = \widehat{A}(t,\varepsilon )z(t)\). This means that \(\varPhi (t,s,\varepsilon )\) satisfies the initial-value problem
Remember that the block matrix \(\widehat{A}(t,\varepsilon )\) is defined in (50)–(51).
Let us partition the matrix \(\varPhi (t,s,\varepsilon )\) into blocks as:
where the matrices \(\varPhi _{1}(t,s,\varepsilon )\), \(\varPhi _{2}(t,s,\varepsilon )\), \(\varPhi _{3}(t,s,\varepsilon )\) and \(\varPhi _{4}(t,s,\varepsilon )\) are of the dimensions \((n-r+q)\times (n-r+q)\), \((n-r+q)\times (r-q)\), \((r-q)\times (n-r+q)\) and \((r-q)\times (r-q)\), respectively.
Along with the problem (73), we consider the following problem with respect to the \((n-r+q)\times (n-r+q)\)-matrix-valued function \(\varPhi _{0}(t,s)\):
where \(A_{0}(t) = \widehat{A}_{1}(t) - \widehat{A}_{2}(t)\widehat{A}_{4}^{-1}(t,0)\widehat{A}_{3}(t,0)\), \(\widehat{A}_{1}(t)\), \(\widehat{A}_{2}(t)\), \(\widehat{A}_{3}(t,\varepsilon )\), \(\widehat{A}_{4}(t,\varepsilon )\) are blocks of of the matrix \(\widehat{A}(t,\varepsilon )\). Since \(\widehat{A}_{4}(t,0)=-P_{3,0}^{o}(t)=-\big (D_{2}(t)\big )^{1/2}\), then \(\widehat{A}_{4}(t,0)\) is invertible. It is clear that the problem (75) has the unique solution \(\varPhi _{0}(t,s)\), \(0\le s \le t \le t_f\).
By virtue of the results of [37] (Lemma 3.1), we have the following assertion.
Lemma 4
Let the assumptions (A1)-(A6), (A8) be valid. Then, there exists a positive number \(\varepsilon _{2}^{*}\) such that, for all \(\varepsilon \in (0,\varepsilon _{2}^{*}]\), the following inequalities are satisfied:
\(\big \Vert \varPhi _{1}(t,s,\varepsilon ) - \varPhi _{0}(t,s)\big \Vert \le a\varepsilon \), \(\big \Vert \varPhi _{2}(t,s,\varepsilon )\big \Vert \le a\varepsilon \), \(\big \Vert \varPhi _{3}(t,s,\varepsilon ) + \widehat{A}_{4}^{-1}(t,0)\widehat{A}_{3}(t,0)\) \(\varPhi _{0}(t,s)\big \Vert \le a\Big [\varepsilon + \exp \Big (- \beta (t-s)/\varepsilon \Big )\Big ]\), \(\big \Vert \varPhi _{4}(t,s,\varepsilon )\big \Vert \le a\Big [\varepsilon + \exp \Big (- \beta (t-s)/\varepsilon \Big )\Big ]\), where \(0\le s \le t \le t_f\); \(a > 0\) and \(\beta > 0\) are some constants independent of \(\varepsilon \).
Main Part of the Proof
Due to the proof of Theorem 2, the vector-valued function \(z_{0}^{*} \big (t,\varepsilon ; w(\cdot )\big )\), \(t\in [0,t_f]\), being the solution of the initial-value problem (7) with \(u(t)=u_{\varepsilon ,0}^{*}[z(t),t]\), also is the solution of the initial-value problem (48).
Since \(\varPhi (t,s,\varepsilon )\) is the fundamental matrix solution of the system \(dz(t)/dt =\) \( \widehat{A}(t,\varepsilon )z(t)\), then the solution of (48) can be represented in the form
Let us partition the vector \(z_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big )\) into blocks as
where \(x_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big )\in E^{n-r+q}\), \(y_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big )\in E^{r-q}\).
Substitution of (24), (74) and (77) into (76) yields after a routine algebra
Using Lemma 4, we obtain the inequalities for all \(0\le s \le t \le t_f\) and \(\varepsilon \in (0,\varepsilon _{2}^{*}]\):
where \(a>0\) is some constant independent of \(\varepsilon \).
Consider the following vector-valued functions of the dimensions \(n-r+q\) and \(r-q\), respectively: \(\varphi _{x}\big (t;w(\cdot )\big )=\int _{0}^{t}\varPhi _{0}(t,s)F_{1}(s)w(s)ds\) and \(\varphi _{y}\big (t;w(\cdot )\big )= \) \(- \widehat{A}_{4}^{-1}(t,0)\widehat{A}_{3}(t,0)\varphi _{x}\big (t;w(\cdot )\big )\), \(t\in [0,t_f]\). We have that
Moreover, using the Eqs. (78)–(79), the inequalities (80)–(81) and the Cauchy-Bunyakovsky-Schwarz integral inequality, we directly obtain the inequalities
where \(a_{1}>0\) is some constant independent of \(\varepsilon \) and \(w(\cdot )\).
Let us denote: \(\Delta x\big (t,\varepsilon ;w(\cdot )\big ) {\mathop {=}\limits ^{\triangle }} x_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big ) - \varphi _{x}\big (t;w(\cdot )\big )\), \(\Delta y\big (t,\varepsilon ;w(\cdot )\big ) {\mathop {=}\limits ^{\triangle }} y_{0}^{*}\big (t,\varepsilon ;w(\cdot )\big ) - \varphi _{y}\big (t;w(\cdot )\big )\). Then, the use of the Eqs. (43), (77), (82) and of the fact that \(\widehat{A}_{3}(t,0)=-\big (P_{2,0}^{o}(t)\big )^{T}\), \(\widehat{A}_{4}(t,0)=-P_{3,0}^{o}(t)\) (see Eq. (51)) yields
The Eq. (84), along with the inequalities (83), yields the inequality
where \(t\in [0,t_f]\); \(\varepsilon \in (0,\varepsilon _{2}^{*}]\); \(a_{2}>0\) is some constant independent of \(\varepsilon \) and \(w(\cdot )\).
The inequality (85) immediately yields the inequality (45), which completes the proof of Theorem 3.
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Glizer, V.Y., Kelis, O. (2020). Finite-Horizon \(H_{\infty }\) Control Problem with Singular Control Cost. In: Gusikhin, O., Madani, K. (eds) Informatics in Control, Automation and Robotics . ICINCO 2017. Lecture Notes in Electrical Engineering, vol 495. Springer, Cham. https://doi.org/10.1007/978-3-030-11292-9_2
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