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.
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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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